Burnham is in the running to become mayor of Manchester and said he was quitting the shadow cabinet in order to focus on the election, which will be held next May.
"This is my tenth Conference speaking to you as a cabinet or shadow cabinet minister," he said as he announced he was standing down as shadow home secretary.
"And it will be my last. It is time for me to turn my full focus to Greater Manchester.
"That's why I can tell you all first today that I have asked Jeremy to plan a new shadow cabinet without me, although I will of course stay until it is in place."
Burnham, who ran against Corbyn in last year's leadership election, said he had been saddened by the events of the past year, and called on his fellow Labour party members to unite.
"It has been my privilege to represent our great party these last 10 years," he told his audience.
"But, at times, the last 12 months have made me profoundly sad. Sad to hear the achievements of our Labour Government, in which I was proud to serve, being dismissed as if they were nothing.
"Sad that old friendships have been strained; sad that some seem to prefer fighting each other than the Tories."
Loyalty to leadership
Burnham continued: "I have given my all to this party and always put its interests above those of factions and personalities. And I have given exactly the same loyalty to all four of the Labour leaders I have served.
Some say that makes me inconsistent. But... I have an old-fashioned belief - that a Labour government of any kind is a million times better for my constituents in Leigh than any Tory government and that is more likely to happen if Labour is united.
"So let us unite - let's have an end to divisive talk about deselections but, in return, let's have more respect for the democratic will of you, our members. This city has reminded us all of the immense power of true solidarity."
Lib Dem reaction
“Only Andy Burnham can resign half way through his own speech," said Lib Dem chief whip Tom Brake.
“I believe in politics there are weathervanes and sign posts. Andy Burnham is the weathervane in a gale. He has had more views on immigration than I’ve had hot dinners.
“For a man who has spent his life facing both ways on this critical issue, his speech today takes the biscuit.”