I appear to be really bad at being annoyed at people on LW without getting downvoted

Don't take it too hard: everyone, everywhere, is really bad at being annoyed at people in general without getting some sort of status hit. This has a lot more to do with the person's reaction to "being annoyed at" and a lot less to do with anyone's skill at "being annoyed at" people.

Well, that wasn't clear at all. There is no safe, good way to be annoyed at people, because people mostly don't like it when someone's annoyed at them. Most of these people will then retaliate in some manner that's supposed to be unpleasant. This is the case online and off.

Sounds profound, but none of that helps you deal with these people when you're annoyed, does it?

I think the answer lies in examining why you want to be annoyed at them: for example, if it's to change their behaviour, then you only need look for more effective ways to change their behaviour than being annoyed. I don't claim that changing their behaviour is your goal in being annoyed; but you should subject whatever reason you do have to some scrutiny and figure out another, potentially more effective way of achieving that goal or goals.