It really irritates me when people are stupid enough not to know the difference between their and they're, your and you're, etc. I mean I have to share a planet with these people, it's painful. It's vs. its bothers me too, but not because it reflects stupidity so much as it reflects the fact that everyone's a member of some kind of groupthink (why else would *everyone* get it wrong all the time?), and that groupthink can be very wrong, with very damaging consequences in the more general case.
I mean some people could argue "I just don't care about the difference between their and they're, you know what I'm saying," but the fact is smart people never get those basic things wrong, and anyway, if it really is about not caring rather than being stupid, that person is anal-expulsive and doesn't give a shit about anything--truth, correctness, integrity, whatever---and that's equally reprehensible.
I'd correct people more often, but I don't want to make enemies and I know how much people hate to be enlightened about their grammar. That irritates me too.
Edit:
It's especially irrational when somebody makes a hypercorrection, for example saying "I felt badly" where it should be "I felt bad" because "felt" acts as a linking verb there, and when you try to correct them they get all offended and lecture you about how language is all made up and the important thing is only that you're understood anyway. The reason this is irrational is because if they're making a hypercorrection, they're actually going out of their way to to try to do the grammar correctly, so they should be appreciative if somebody tells you how to actually do it correctly. In the case of "I felt badly," it's obvious that they're trying really hard to be correct because they're forcing the "badly" to agree with "felt" because it looks like an adverb, despite how uncommon and unsightly saying it that way is.