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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2021-05-27</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0001847874</issuerCik>
        <issuerName>Fifth Wall Acquisition Corp. III</issuerName>
        <issuerTradingSymbol>FWAC</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001847658</rptOwnerCik>
            <rptOwnerName>Fifth Wall Acquisition Sponsor III, LLC</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>6060 CENTER DRIVE</rptOwnerStreet1>
            <rptOwnerStreet2>10TH FLOOR</rptOwnerStreet2>
            <rptOwnerCity>LOS ANGELES</rptOwnerCity>
            <rptOwnerState>CA</rptOwnerState>
            <rptOwnerZipCode>90045</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>0</isDirector>
            <isOfficer>0</isOfficer>
            <isTenPercentOwner>1</isTenPercentOwner>
            <isOther>0</isOther>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Class A Ordinary Shares</value>
            </securityTitle>
            <transactionDate>
                <value>2021-05-27</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>P</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionTimeliness>
                <value></value>
            </transactionTimeliness>
            <transactionAmounts>
                <transactionShares>
                    <value>907000</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>10.00</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>907000</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                    <footnoteId id="F2"/>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <footnotes>
        <footnote id="F1">Represents the purchase of shares of Class A Ordinary Shares of the Issuer in a private placement in connection with Issuer's initial public offering.</footnote>
        <footnote id="F2">Fifth Wall Acquisition Sponsor III LLC, is the record holder of the shares reported herein. Fifth Wall Acquisition Sponsor III LLC is governed by two managers. Messr. Andriy Mykhaylovskyy and Brendan Wallace have shared voting and investment power over the shares held by Fifth Wall Acquisition Sponsor III LLC, and disclaim beneficial ownership of the shares held by Fifth Wall Acquisition Sponsor III LLC except to the extent of their pecuniary interest therein.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>/s/ Andriy Mykhaylovskyy, Manager</signatureName>
        <signatureDate>2021-05-27</signatureDate>
    </ownerSignature>
</ownershipDocument>
