<?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>0001844111</rptOwnerCik>
            <rptOwnerName>Wallace Brendan</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>C/O FIFTH WALL ACQUISITION CORP. III</rptOwnerStreet1>
            <rptOwnerStreet2>6060 CENTER DRIVE, 10TH FLOOR</rptOwnerStreet2>
            <rptOwnerCity>LOS ANGELES</rptOwnerCity>
            <rptOwnerState>CA</rptOwnerState>
            <rptOwnerZipCode>90045</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>1</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>Chairman of the Board and CEO</officerTitle>
        </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>I</value>
                </directOrIndirectOwnership>
                <natureOfOwnership>
                    <value>See Footnote</value>
                    <footnoteId id="F2"/>
                </natureOfOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <footnotes>
        <footnote id="F1">Represents the purchase of shares of Class A Ordinary Shares of the Issuer by Fifth Wall Acquisition Sponsor III LLC 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/ Brendan Wallace</signatureName>
        <signatureDate>2021-05-27</signatureDate>
    </ownerSignature>
</ownershipDocument>
