Appearing on the edition with the prime minister will be comedian Joe Lycett and Sugababes singers Mutya Buena, Keisha Buchanan and Siobhan Donaghy.
A Channel 4 spokesperson said nothing would be “off limits as Gogglebox continues to celebrate the simple joy of watching television together”.
“They’ll be sharing hilarious, honest and often unexpected reactions to the week’s biggest television moments as well as poignant stories all in aid of Stand Up To Cancer,” they added.
Burnham and Rotheram have been friends for many years, serving together as Labour MPs before they both left Parliament to become regional mayors in 2017.
They remained close allies, writing a book together on their vision for Britain’s future, and Rotheram was a visible supporter of Burnham’s successful campaign to become the MP for Makerfield in June.
The Liverpool City Region mayor said while filming their contributions, he and Burnham had struggled with one particularly cerebral TV quiz show and found themselves flummoxed by its music round.
“We’re both big into our music but we couldn’t get it,” he said.
He added that while their television tastes were largely similar, he and the prime minister disagreed over what snacks to eat while watching.
“I’m more of a cheesy puffs-type guy, but he’s more into the hard, crunchy stuff,” he said.