Tom Harris, who stood at the Scottish Labour Party's leadership race last year, apologised for posting the video, which took footage from 2004 film Downfall but replaced the Nazi leader's voice with subtitled words of SNP leader Salmond.
The Downfall parody clip has also been widely used to mock other politicians, including former prime minister Gordon Brown.
However, the SNP said that the video was "tasteless" and claimed that the situation was "hugely embarrassing" for the Labour party.
Harris was put in charge of the Scottish Labour Party's new media strategy after previous incumbent Johann Lamont was elected party leader.
He posted a version of the video, entitled Joan's Downfall, on his own website in reference to SNP MSP Joan McAlpine's claim that Labour and the Lib Dems were being "anti-Scottish" in their stance on a proposed independence referendum.
The Labour MP for Glasgow South said in a statement: "Having spoken to Johann, I have decided to step down from leading the party's social media review. The video I posted has been a well-worn joke used to parody a range of public figures.
"However, context is everything and in the context of Johann's and my desire to improve the level of political debate on social media and the context of Joan McAlpine's much more serious statements about all political opponents of the SNP being anti-Scottish, my actions have been an unhelpful distraction, for which I apologise."
Speaking about the video, an SNP spokesperson said: "It is silly, negative nonsense like this that helps explain why Labour are in the doldrums in Scotland.
"It is hugely embarrassing for Labour that one of their MPs and a leadership contender - who should have better things to do with his time - is promoting tasteless stuff like this through his official MP website."
A Scottish Labour spokesperson said that attention should now be on having "an adult debate" about the future of Scotland.
The spokesperson added: "Attention is now focusing again on the SNP's refusal to take any action against their official ministerial aide Joan McAlpine for outrageously questioning people's Scottishness and patriotism if, like most people, they don't support separation."
It has been a difficult start to 2012 for the Labour Party digitally, after MP Diane Abbott was recently embroiled in a racism row over comments she made about "white people" on Twitter.
Watch Joan's Downfall below: