Hmm, I have several. I'll put up with a lot if I love the idea or character(s) enough, but these things bug me:
1) Using a story as a way to bash a character one doesn't like. I've been in fandom since 1997, and I've seen it in fandoms from X-Files to Twilight. If the writer doesn't like ___ character,
just avoid writing him/her. Bashing a character in fanfiction is bad characterization and THAT bugs me.
2) Failing to look things up. "Practice the art of getting it right." There IS, I think, a "sliding scale" of the import of details. A minor detail in a story that happens to be wrong will bug me far less than a key element on which the entire story turns. I know that some people use, "But it's just fanfiction!" as an excuse, but I don't, myself, consider it one. Sometimes a simple Google search can net one the answer. (This goes for canon details, for RL details, historical details ... it all pertains.)
3) Failing to proofread, or get a beta reader. Again, a few typos or occasional problems is NOT going to throw me off a story. But extremely poor grammar, bad spelling ... yes, that will.
4) Failing to finish a story that one began ... especially without explanation. Sometimes RL intervenes; I understand that. It's happened to me. But to drop a story without warning or explanation is rude to readers. Sometimes writers write themselves into a corner -- that can be avoided by
plotting ahead of time.
Those are the big ones, I suppose. I do my absolute best to avoid these things myself, so I do practice what I preach.