The footage also showed her questioning why Holocaust Memorial Day was not more wide-ranging to include other genocides, when in fact it does include all post- World War II genocides.
Labour said it did not comment on individual party memberships but it is understood Ms Walker has now been suspended, The Press Association reported.
Labour MP John Mann called Ms Walker’s comments “unacceptable in a modern political party” by any standard.
He also suggested the comments had “inspired waves of anti-Semitic and racist backlash including Holocaust denial”.
But former London mayor Ken Livingstone defended some of Ms Walker’s comments, saying “there’s a difference between ignorance and anti-Semitism”.
Mr Mann said: “Enough is enough. Though she claims impunity for many reasons, Jackie Walker’s behaviour is discriminatory, provocative, offensive and by any standard unacceptable in a modern political party.
“Not only has she caused offence personally, she has inspired waves of anti-Semitic and racist backlash including Holocaust denial.
“Not only must she be expelled from the Labour Party immediately but all those abusing others in supporting her must go too.
“Temporary suspensions are not good enough, these people must be given permanent bans and no platform to express their anti-Semitism anywhere in the Labour Party, if we are to be serious about opposing anti-Jewish hatred.”
Mr Livingstone, though, said most people were unaware that Holocaust Memorial Day commemorated other genocides besides those perpetrated by the Nazis.
“I suspect you’ll find the majority of people in Britain didn’t know the Holocaust Memorial Day had been widened to include others,” Mr Livingstone told the Press Association.
“There’s a difference between ignorance and anti-Semitism.”
Ms Walker, who says she and her partner are Jewish, previously released a statement apologising for any offence.
In an interview with Channel 4 News, she also questioned why the Holocaust only marked genocides that happened after the Nazis.
When she was asked whether she had considered resigning given the outrage among some Jewish groups, Ms Walker said: “Some other prominent Jewish groups, of which I’m a member of, think a very different thing.
“What we have to look at when we’re talking about this subject, particularly at the moment, is the political differences that are underlying this as well.”
Ms Walker said whoever leaked the footage from a Labour Party anti-Semitism training event “had malicious intent in their mind”.
She also said she was anti-Zionist rather than anti-Semitic, adding: “I think Zionism is a political ideology, and like any political ideology, some people will be supportive and some people won’t be supportive of it. That’s a very different thing.”
The steering group of Momentum, which rose from the campaign to get Mr Corbyn elected as Labour leader last year, meets on Monday.
A spokesman for the group said that it would be seeking to remove Ms Walker as its vice-chairwoman.
Ms Walker was previously suspended from the Labour Party for comments on social media saying Jews were the “chief financiers of the sugar and slave trade”.
She was readmitted to the party after an investigation.