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

    <schemaVersion>X0306</schemaVersion>

    <documentType>4</documentType>

    <periodOfReport>2016-02-02</periodOfReport>

    <notSubjectToSection16>0</notSubjectToSection16>

    <issuer>
        <issuerCik>0000949870</issuerCik>
        <issuerName>BOSTON BEER CO INC</issuerName>
        <issuerTradingSymbol>SAM</issuerTradingSymbol>
    </issuer>

    <reportingOwner>
        <reportingOwnerId>
            <rptOwnerCik>0001206513</rptOwnerCik>
            <rptOwnerName>ROPER MARTIN F</rptOwnerName>
        </reportingOwnerId>
        <reportingOwnerAddress>
            <rptOwnerStreet1>C/O THE BOSTON BEER COMPANY, INC.</rptOwnerStreet1>
            <rptOwnerStreet2>ONE DESIGN CENTER PLACE, SUITE 850</rptOwnerStreet2>
            <rptOwnerCity>BOSTON</rptOwnerCity>
            <rptOwnerState>MA</rptOwnerState>
            <rptOwnerZipCode>02210</rptOwnerZipCode>
            <rptOwnerStateDescription></rptOwnerStateDescription>
        </reportingOwnerAddress>
        <reportingOwnerRelationship>
            <isDirector>1</isDirector>
            <isOfficer>1</isOfficer>
            <isTenPercentOwner>0</isTenPercentOwner>
            <isOther>0</isOther>
            <officerTitle>President and C.E.O.</officerTitle>
            <otherText></otherText>
        </reportingOwnerRelationship>
    </reportingOwner>

    <nonDerivativeTable>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Class A Common</value>
            </securityTitle>
            <transactionDate>
                <value>2016-02-02</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>M</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>10000</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>105.33</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>A</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>12273</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Class A Common</value>
            </securityTitle>
            <transactionDate>
                <value>2016-02-02</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>S</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>3272</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>174.50</value>
                    <footnoteId id="F2"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>9001</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Class A Common</value>
            </securityTitle>
            <transactionDate>
                <value>2016-02-02</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>S</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>4893</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>175.43</value>
                    <footnoteId id="F3"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>4108</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Class A Common</value>
            </securityTitle>
            <transactionDate>
                <value>2016-02-02</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>S</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>2700</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>176.16</value>
                    <footnoteId id="F4"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>2700</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Class A Common</value>
            </securityTitle>
            <transactionDate>
                <value>2016-02-02</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>S</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>227</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>177.00</value>
                    <footnoteId id="F5"/>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>2473</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
        <nonDerivativeTransaction>
            <securityTitle>
                <value>Class A Common</value>
            </securityTitle>
            <transactionDate>
                <value>2016-02-02</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>S</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>200</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>178.64</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>2273</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </nonDerivativeTransaction>
    </nonDerivativeTable>

    <derivativeTable>
        <derivativeTransaction>
            <securityTitle>
                <value>Class A Common Stock Option</value>
            </securityTitle>
            <conversionOrExercisePrice>
                <value>105.33</value>
            </conversionOrExercisePrice>
            <transactionDate>
                <value>2016-02-02</value>
            </transactionDate>
            <deemedExecutionDate></deemedExecutionDate>
            <transactionCoding>
                <transactionFormType>4</transactionFormType>
                <transactionCode>M</transactionCode>
                <equitySwapInvolved>0</equitySwapInvolved>
                <footnoteId id="F1"/>
            </transactionCoding>
            <transactionAmounts>
                <transactionShares>
                    <value>10000</value>
                </transactionShares>
                <transactionPricePerShare>
                    <value>0</value>
                </transactionPricePerShare>
                <transactionAcquiredDisposedCode>
                    <value>D</value>
                </transactionAcquiredDisposedCode>
            </transactionAmounts>
            <exerciseDate>
                <value>2014-01-01</value>
                <footnoteId id="F6"/>
            </exerciseDate>
            <expirationDate>
                <value>2018-12-31</value>
                <footnoteId id="F6"/>
            </expirationDate>
            <underlyingSecurity>
                <underlyingSecurityTitle>
                    <value>Class A Common</value>
                </underlyingSecurityTitle>
                <underlyingSecurityShares>
                    <value>753864</value>
                </underlyingSecurityShares>
            </underlyingSecurity>
            <postTransactionAmounts>
                <sharesOwnedFollowingTransaction>
                    <value>342319</value>
                </sharesOwnedFollowingTransaction>
            </postTransactionAmounts>
            <ownershipNature>
                <directOrIndirectOwnership>
                    <value>D</value>
                </directOrIndirectOwnership>
            </ownershipNature>
        </derivativeTransaction>
    </derivativeTable>

    <footnotes>
        <footnote id="F1">The transactions reported in this Form 4 were effected pursuant to a Rule 10b5-1 trading plan adopted by the Reporting Person on November 9, 2015.</footnote>
        <footnote id="F2">The price shown is the weighted average sale price for the transactions reported on this line. The range of sale prices for the 3,272 shares is from $173.93 to $174.92. The Filing Person will provide full information regarding the number of shares sold at each separate price upon request of the SEC, the Registrant, or a shareholder of the Registrant.</footnote>
        <footnote id="F3">The price shown is the weighted average sale price for the transactions reported on this line. The range of sale prices for the 4,893 shares is from $174.93 to $175.86. The Filing Person will provide full information regarding the number of shares sold at each separate price upon request of the SEC, the Registrant, or a shareholder of the Registrant.</footnote>
        <footnote id="F4">The price shown is the weighted average sale price for the transactions reported on this line. The range of sale prices for the 1,408 shares is from $175.93  to $176.84. The Filing Person will provide full information regarding the number of shares sold at each separate price upon request of the SEC, the Registrant, or a shareholder of the Registrant.</footnote>
        <footnote id="F5">The price shown is the weighted average sale price for the transactions reported on this line. The range of sale prices for the 227 shares is from $177.00 to $177.02. The Filing Person will provide full information regarding the number of shares sold at each separate price upon request of the SEC, the Registrant, or a shareholder of the Registrant.</footnote>
        <footnote id="F6">The option vests in five equal installments; the first on January 1, 2014, and the final vesting on January 1, 2018, subject to the Reporting Person remaining employed by the Issuer on the applicable vesting date. With respect to certain shares, the option expires on December 31, 2017. All options are subject to earlier expiration based on certain contingencies.</footnote>
    </footnotes>

    <remarks></remarks>

    <ownerSignature>
        <signatureName>Kathleen H. Wade under POA for the benefit of Martin F. Roper</signatureName>
        <signatureDate>2016-02-03</signatureDate>
    </ownerSignature>
</ownershipDocument>
