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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4/A</documentType>

    <periodOfReport>2013-11-18</periodOfReport>

    <dateOfOriginalSubmission>2013-11-20</dateOfOriginalSubmission>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0001415624</issuerCik>
        <issuerName>YuMe Inc</issuerName>
        <issuerTradingSymbol>YUME</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001189229</rptOwnerCik>
            <rptOwnerName>PAISLEY CHRISTOPHER B</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>C/O YUME, INC.</rptOwnerStreet1>
            <rptOwnerStreet2>1204 MIDDLEFIELD ROAD</rptOwnerStreet2>
            <rptOwnerCity>REDWOOD CITY</rptOwnerCity>
            <rptOwnerState>CA</rptOwnerState>
            <rptOwnerZipCode>94063</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>2013-11-18</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>P</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>10000</value>
                    <footnoteId id="F1"/>
                </transactionShares>
                <transactionPricePerShare>
                    <value>9.2357</value>
                    <footnoteId id="F2"/>
                    <footnoteId id="F3"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>10000</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <derivativeTable></derivativeTable>

    <footnotes>
        <footnote id="F1">Represents the aggregate of purchases effected on the same day at different prices.</footnote>
        <footnote id="F2">Represents the weighted average purchase price per share. The shares were purchased at prices ranging from $9.17 to $9.25 per share. Full information regarding the number of shares purchased at each price shall be provided upon request from the staff of the U.S. Securities and Exchange Commission, the Issuer, or a security holder of the Issuer.</footnote>
        <footnote id="F3">This Form 4/A amends the purchase price of the shares purchased, which had been erroneously reported on a Form 4 filed on November 20, 2013.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/s/ Christopher B. Paisley by Jeremy Ford, Attorney-in-Fact</signatureName>
        <signatureDate>2013-11-25</signatureDate>
    </ownerSignature>
</ownershipDocument>
