The former sports minister told BBC Radio 5 Live's Pienaar's Politics there was a "pretty overwhelming case" for the Football Association to act.
He said he had long had doubts about Russia hosting the event.
Head of the FA, Greg Dyke was abit more cautious though. He told BBC One's the Andrew Marr Show: "It's got to be done by enough nations to have an impact, if it's done.
"It would be ridiculous to try to do it on your own because all we'd do is pull out of the World Cup and everyone would say 'well done' and they'd forget all about us.
"We've got to do it alongside other large footballing nations."
Mr Burnham said: "I think the FA should be a bit stronger personally and should say what many people feel, that it's just not acceptable, particularly when you look at the alleged corruption within Fifa and then the situation between Russia and Ukraine.
"It would be very difficult but sometimes you've got to take a stand and you've got to say whether it's right or wrong.
"And I can't see how it feels right to me to send an England team to Russia to play in a World Cup as if nothing has happened. Qatar too - but that's an issue that's further down the line.
"If enough people take that stand and follow our lead, then we will see new arrangements for the 2018 World Cup which I believe is what we need."