<?xml version="1.0"?>
<ownershipDocument>

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2021-08-18</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0000096943</issuerCik>
        <issuerName>TELEFLEX INC</issuerName>
        <issuerTradingSymbol>TFX</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001552476</rptOwnerCik>
            <rptOwnerName>Kelly Liam</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>550 E. SWEDESFORD ROAD</rptOwnerStreet1>
            <rptOwnerStreet2></rptOwnerStreet2>
            <rptOwnerCity>WAYNE</rptOwnerCity>
            <rptOwnerState>PA</rptOwnerState>
            <rptOwnerZipCode>19087</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>1</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>Chairman, President &amp; CEO</officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2021-08-18</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>M</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>8987</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>59.75</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>34762.491</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2021-08-18</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>S</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>1007</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>374.85</value>
                    <footnoteId id="F2"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>33755.491</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2021-08-18</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>S</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>4045</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>375.32</value>
                    <footnoteId id="F3"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>29710.491</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2021-08-18</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>S</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>1051</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>376.43</value>
                    <footnoteId id="F4"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>28659.491</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2021-08-18</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>S</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>1513</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>377.53</value>
                    <footnoteId id="F5"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>27146.491</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2021-08-18</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>S</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>1371</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>378.92</value>
                    <footnoteId id="F6"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>25775.491</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <derivativeTable>
        <derivativeTransaction>
            <securityTitle>
                <value>Stock Option / (Right to Buy)</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>59.75</value>
            </conversionOrExercisePrice>
            <transactionDate>
                <value>2021-08-18</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>M</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>8987</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>59.75</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <exerciseDate>
                <footnoteId id="F7"/>
            </exerciseDate>
            <expirationDate>
                <value>2022-02-28</value>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>8987.0</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>2276</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeTransaction>
    </derivativeTable>

    <footnotes>
        <footnote id="F1">Transaction effected pursuant to a Rule 10b5-1 trading plan established by the reporting person on May 21, 2021.</footnote>
        <footnote id="F2">This transaction was executed in multiple trades at prices ranging from $374.02 to $375.02. The price reported reflects the weighted average sale price. The reporting person hereby undertakes to provide, upon request, to the SEC staff, the issuer or a security holder of the issuer full information regarding the number of shares and prices at which the transaction was effected.</footnote>
        <footnote id="F3">This transaction was executed in multiple trades at prices ranging from $375.03 to $376.03. The price reported reflects the weighted average sale price. The reporting person hereby undertakes to provide, upon request, to the SEC staff, the issuer or a security holder of the issuer full information regarding the number of shares and prices at which the transaction was effected.</footnote>
        <footnote id="F4">This transaction was executed in multiple trades at prices ranging from $376.05 to $377.02. The price reported reflects the weighted average sale price. The reporting person hereby undertakes to provide, upon request, to the SEC staff, the issuer or a security holder of the issuer full information regarding the number of shares and prices at which the transaction was effected.</footnote>
        <footnote id="F5">This transaction was executed in multiple trades at prices ranging from $377.06 to $377.98. The price reported reflects the weighted average sale price. The reporting person hereby undertakes to provide, upon request, to the SEC staff, the issuer or a security holder of the issuer full information regarding the number of shares and prices at which the transaction was effected.</footnote>
        <footnote id="F6">This transaction was executed in multiple trades at prices ranging from $378.195 to $379.15. The price reported reflects the weighted average sale price. The reporting person hereby undertakes to provide, upon request, to the SEC staff, the issuer or a security holder of the issuer full information regarding the number of shares and prices at which the transaction was effected.</footnote>
        <footnote id="F7">Exercisable for one-third of the shares on each of 2/28/2013, 2/28/2014 and 2/28/2015.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>Daniel V. Logue with POA for Liam Kelly</signatureName>
        <signatureDate>2021-08-20</signatureDate>
    </ownerSignature>
</ownershipDocument>
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X0306
4
2021-08-18
0
0000096943
TELEFLEX INC
TFX
0001552476
Kelly Liam
550 E. SWEDESFORD ROAD
WAYNE
PA
19087
1
1
0
0
Chairman, President & CEO
Common Stock
2021-08-18
4
M
0
8987
59.75
A
34762.491
D
Common Stock
2021-08-18
4
S
0
1007
374.85
D
33755.491
D
Common Stock
2021-08-18
4
S
0
4045
375.32
D
29710.491
D
Common Stock
2021-08-18
4
S
0
1051
376.43
D
28659.491
D
Common Stock
2021-08-18
4
S
0
1513
377.53
D
27146.491
D
Common Stock
2021-08-18
4
S
0
1371
378.92
D
25775.491
D
Stock Option / (Right to Buy)
59.75
2021-08-18
4
M
0
8987
59.75
D
2022-02-28
Common Stock
8987.0
2276
D
Transaction effected pursuant to a Rule 10b5-1 trading plan established by the reporting person on May 21, 2021.
This transaction was executed in multiple trades at prices ranging from $374.02 to $375.02. The price reported reflects the weighted average sale price. The reporting person hereby undertakes to provide, upon request, to the SEC staff, the issuer or a security holder of the issuer full information regarding the number of shares and prices at which the transaction was effected.
This transaction was executed in multiple trades at prices ranging from $375.03 to $376.03. The price reported reflects the weighted average sale price. The reporting person hereby undertakes to provide, upon request, to the SEC staff, the issuer or a security holder of the issuer full information regarding the number of shares and prices at which the transaction was effected.
This transaction was executed in multiple trades at prices ranging from $376.05 to $377.02. The price reported reflects the weighted average sale price. The reporting person hereby undertakes to provide, upon request, to the SEC staff, the issuer or a security holder of the issuer full information regarding the number of shares and prices at which the transaction was effected.
This transaction was executed in multiple trades at prices ranging from $377.06 to $377.98. The price reported reflects the weighted average sale price. The reporting person hereby undertakes to provide, upon request, to the SEC staff, the issuer or a security holder of the issuer full information regarding the number of shares and prices at which the transaction was effected.
This transaction was executed in multiple trades at prices ranging from $378.195 to $379.15. The price reported reflects the weighted average sale price. The reporting person hereby undertakes to provide, upon request, to the SEC staff, the issuer or a security holder of the issuer full information regarding the number of shares and prices at which the transaction was effected.
Exercisable for one-third of the shares on each of 2/28/2013, 2/28/2014 and 2/28/2015.
Daniel V. Logue with POA for Liam Kelly
2021-08-20