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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2019-01-03</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0001171759</issuerCik>
        <issuerName>RED ROBIN GOURMET BURGERS INC</issuerName>
        <issuerTradingSymbol>RRGB</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001354064</rptOwnerCik>
            <rptOwnerName>Constant Guy J</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>6312 S FIDDLER'S GREEN CIR</rptOwnerStreet1>
            <rptOwnerStreet2>SUITE 200N</rptOwnerStreet2>
            <rptOwnerCity>GREENWOOD VILLAGE</rptOwnerCity>
            <rptOwnerState>CO</rptOwnerState>
            <rptOwnerZipCode>80111</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>Chief Financial Officer</officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2019-01-03</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>F</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>314</value>
                    <footnoteId id="F1"/>
                </transactionShares>
                <transactionPricePerShare>
                    <value>27.20</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>36053</value>
                    <footnoteId id="F2"/>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <derivativeTable></derivativeTable>

    <footnotes>
        <footnote id="F1">In connection with the vesting of 917 time-based restricted stock units which were granted on January 3, 2017 under the issuer's Second Amended and Restated 2007 Performance Incentive Plan, the reporting person sold 314 of such shares to the issuer to satisfy tax withholding obligations. The sale was approved by the issuer's Compensation Committee in accordance with Rule 16b-3(d)(1) of the Exchange Act (the &quot;Act&quot;), and as such, the sale is exempt from Section 16(b) of the Act pursuant to Rule 16b-3(e) promulgated thereunder.</footnote>
        <footnote id="F2">Includes 22,374 shares subject to vesting and forfeiture restrictions</footnote>
    </footnotes>

    <ownerSignature>
        <signatureName>/s/ Michael L. Kaplan, Attorney-in-Fact</signatureName>
        <signatureDate>2019-01-07</signatureDate>
    </ownerSignature>
</ownershipDocument>
