-Study your homework or not (don't) , it is your problem.
-Send me a post card or not (don't).
-Go or not (don't).
I can't figure out which one is grammatically OK in these sentences, is it with not or don't?
In the case of these imperatives, "not" is not appropriate, but "don't" is insufficient on its own. I would express these as:
Study your homework, or don't study your homework [or don't bother]; it is your problem.
Send me a post card, or don't bother.
Go, or don't go.
I'm not a teacher of English, but I have spoken it for (almost) all of my life....