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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2022-05-03</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

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

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001783908</rptOwnerCik>
            <rptOwnerName>Patil Neena M</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>2005 MARKET STREET</rptOwnerStreet1>
            <rptOwnerStreet2>SUITE 2100</rptOwnerStreet2>
            <rptOwnerCity>PHILADELPHIA</rptOwnerCity>
            <rptOwnerState>PA</rptOwnerState>
            <rptOwnerZipCode>19103</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>1</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle></officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2022-05-03</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>A</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>399</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>0</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>399</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>284.85</value>
            </conversionOrExercisePrice>
            <transactionDate>
                <value>2022-05-03</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>A</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>1937</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>0</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <exerciseDate>
                <value>2022-11-03</value>
                <footnoteId id="F2"/>
            </exerciseDate>
            <expirationDate>
                <value>2032-05-03</value>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Common Stock</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>1937.0</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>1937</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeTransaction>
    </derivativeTable>

    <footnotes>
        <footnote id="F1">Restricted Stock Unit Award granted pursuant to Teleflex Incorporated 2014 Stock Incentive Plan. Award vests on November 3, 2022.</footnote>
        <footnote id="F2">Stock Option Award granted pursuant to Teleflex Incorporated 2014 Stock Incentive Plan. Award vests on November 3, 2022.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>Daniel V. Logue with POA for Neena M. Patil</signatureName>
        <signatureDate>2022-05-04</signatureDate>
    </ownerSignature>
</ownershipDocument>
[{"type": "4", "sequence": "1", "filename": "wf-form4_165175899480577.xml", "description": "FORM 4", "secsgml_size_bytes": 5447}]
X0306
4
2022-05-03
0
0000096943
TELEFLEX INC
TFX
0001783908
Patil Neena M
2005 MARKET STREET
SUITE 2100
PHILADELPHIA
PA
19103
1
0
0
0
Common Stock
2022-05-03
4
A
0
399
0
A
399
D
Stock Option / (Right to Buy)
284.85
2022-05-03
4
A
0
1937
0
A
2022-11-03
2032-05-03
Common Stock
1937.0
1937
D
Restricted Stock Unit Award granted pursuant to Teleflex Incorporated 2014 Stock Incentive Plan. Award vests on November 3, 2022.
Stock Option Award granted pursuant to Teleflex Incorporated 2014 Stock Incentive Plan. Award vests on November 3, 2022.
Daniel V. Logue with POA for Neena M. Patil
2022-05-04