Green’s Dictionary of Slang

cheese off v.¹

to annoy.

1950		K. Amis letter 30 July in Leader (2000) 237: Another thing that has been cheesing me off is that Margaret has thrown David over in favour of some rich gracious-living unsuccessful Tory candidate.
1959		G. Slatter Gun in My Hand 200: Cheeses me off, that sort of thing, people going on like animals.
2004		(con. 1960s-70s) T. Taylor Top Fellas 72/2: Which as a fan, and as a Aussie, cheeses me off.