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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2018-11-09</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

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

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001242740</rptOwnerCik>
            <rptOwnerName>POWELL THOMAS E</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>550 E SWEDESFORD ROAD</rptOwnerStreet1>
            <rptOwnerStreet2>SUITE 400</rptOwnerStreet2>
            <rptOwnerCity>WAYNE</rptOwnerCity>
            <rptOwnerState>PA</rptOwnerState>
            <rptOwnerZipCode>19087</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>Executive Vice President &amp; CFO</officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2018-11-09</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>M</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>5200</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>78.62</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>15351</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2018-11-09</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>S</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>700</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>263.19</value>
                    <footnoteId id="F2"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>14651</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2018-11-09</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>S</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>1800</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>264.28</value>
                    <footnoteId id="F3"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>12851</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2018-11-09</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>S</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>2200</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>265.41</value>
                    <footnoteId id="F4"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>10651</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2018-11-09</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>S</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>500</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>266.45</value>
                    <footnoteId id="F5"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>10151</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>78.62</value>
            </conversionOrExercisePrice>
            <transactionDate>
                <value>2018-11-09</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>M</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>5200</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>0</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <exerciseDate>
                <footnoteId id="F6"/>
            </exerciseDate>
            <expirationDate>
                <value>2023-02-26</value>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>5200.0</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>15824</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, 2018.</footnote>
        <footnote id="F2">This transaction was executed in multiple trades at prices ranging from $262.74 to $263.385. The price reported reflects the weighted average purchase 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 $263.88 to $264.635. The price reported reflects the weighted average purchase 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 $264.92 to $265.78. The price reported reflects the weighted average purchase 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 $266.01 to $266.83. The price reported reflects the weighted average purchase 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">Exercisable for one-third of the shares on each of 2/26/2014, 2/26/2015 and 2/26/2016.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>Daniel V. Logue with POA for Thomas E. Powell</signatureName>
        <signatureDate>2018-11-12</signatureDate>
    </ownerSignature>
</ownershipDocument>
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X0306
4
2018-11-09
0
0000096943
TELEFLEX INC
TFX
0001242740
POWELL THOMAS E
550 E SWEDESFORD ROAD
SUITE 400
WAYNE
PA
19087
0
1
0
0
Executive Vice President & CFO
Common Stock
2018-11-09
4
M
0
5200
78.62
A
15351
D
Common Stock
2018-11-09
4
S
0
700
263.19
D
14651
D
Common Stock
2018-11-09
4
S
0
1800
264.28
D
12851
D
Common Stock
2018-11-09
4
S
0
2200
265.41
D
10651
D
Common Stock
2018-11-09
4
S
0
500
266.45
D
10151
D
Stock Option / (Right to Buy)
78.62
2018-11-09
4
M
0
5200
0
D
2023-02-26
Common Stock
5200.0
15824
D
Transaction effected pursuant to a Rule 10b5-1 trading plan established by the reporting person on May 21, 2018.
This transaction was executed in multiple trades at prices ranging from $262.74 to $263.385. The price reported reflects the weighted average purchase 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 $263.88 to $264.635. The price reported reflects the weighted average purchase 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 $264.92 to $265.78. The price reported reflects the weighted average purchase 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 $266.01 to $266.83. The price reported reflects the weighted average purchase 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 2/26/2014, 2/26/2015 and 2/26/2016.
Daniel V. Logue with POA for Thomas E. Powell
2018-11-12