The frontbencher, an ally of Jeremy Corbyn, denied making the remarks when challenged on TV, insisting that "was not my view" and "I didn't make those comments".
But footage has now emerged showing Mr Burgon saying: "Zionism is the enemy of peace and the enemy of the Palestinian people."
The Leeds East MP faced calls to apologise from the Board of Deputies of British Jews.
Amanda Bowman, vice president of the Board of Deputies, said: "These comments were shameful.
"Richard Burgon's denial and the subsequent revelation of his 2014 incitement against Zionists encapsulate the total sham of Labour's approach to anti-Semitism.
"At the very least he should apologise for his comments and for his denial of them.
"The Jewish community has been consistently gaslighted by the Labour Party and they continue to abdicate their responsibility to deal with anti-Semitism in their ranks."
The footage of Mr Burgon's comments, from 2014, was unearthed by investigative reporter Iggy Ostanin.
Mr Burgon is shown saying: "The enemy of the Palestinian people is not the Jewish people, the enemy of the Palestinian people are Zionists and Zionism is the enemy of peace and the enemy of the Palestinian people.
"We need to be loud, we need to be proud in support of a free Palestine."
Jewish Leadership Council chairman Jonathan Goldstein challenged Mr Burgon to issue a "full apology and a clarification as well as a pledge to use responsible language henceforth".
Andrew Neil, the BBC journalist who challenged Mr Burgon last year following reports of his comments, said: "Wish I'd had that clip from 2014 when I interviewed Mr Burgon."
Challenged about the remarks by Neil in March 2018, Mr Burgon said: "I didn't make those comments."
Mr Burgon said in a statement today: "When it was put to me in August 2016 that I had made these remarks I did not recall doing so and therefore asked for the full quotes to be provided to me, and asked when and where I had said it.
"I received no reply, so I believed it was inaccurate to have claimed that I had used that phrase. It is now clear that I did and I regret doing so."