"That's okay" to mean "no" or "don't bother"?

I don’t think it’s a regionalism. I’m from Tennessee and it sounds perfectly natural to me. And there’s evidence from the movies:

Nicky: [after being sick in Andy's face] I'm sorry. I'll still have sex with you if you want.
Andy Stitzer: That's OK.
—The 40-Year-Old Virgin

(Of course, real people don’t talk like movie characters, but here the joke only works if the audience understands “That’s OK” the way you and I do.)

Apparently Canadians say it too:

Bob: Sorry I ralphed, Pam.
Pam: You shouldn't have had that chocolate milk.
Bob: I know. Sorry about your sweater. You can take it out of my pay if you want.
Pam: That's OK.
Bob: Geez, you're real nice. If I didn't have puke breath I'd kiss you.
—Strange Brew

It might be a North Americanism, I guess, but I suspect it’s not even that, just the sort of counterintuitive idiom non-native speakers naturally have trouble with.

I use this construction, and the related "I'm good", and I'm from England. I have friends who also use the same constructions. While I wouldn't say that it's a universal usage, I would expect people here to understand it.