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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2020-02-24</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

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

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001587883</rptOwnerCik>
            <rptOwnerName>Kaplan Michael L.</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>6312 S. FIDDLER'S GREEN CIRCLE</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 Legal Officer</officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Common Stock</value>
            </securityTitle>
            <transactionDate>
                <value>2020-02-24</value>
            </transactionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>F</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>113</value>
                    <footnoteId id="F1"/>
                </transactionShares>
                <transactionPricePerShare>
                    <value>35.18</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>8118</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 330 time-based restricted stock units which were granted on February 24, 2017 under the issuer's Second Amended and Restated 2007 Performance Incentive Plan, the reporting person sold 113 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 4,821 restricted stock units subject to vesting and forfeiture restrictions.</footnote>
    </footnotes>

    <ownerSignature>
        <signatureName>/s/ Michael L. Kaplan</signatureName>
        <signatureDate>2020-02-25</signatureDate>
    </ownerSignature>
</ownershipDocument>
