Not really a fandom/fanbase, but the people on TF2 who aren't free-to-play and mock the people who are...
imo, I do find F2P players frustrating. There may be some really skilled ones who never went p2p, but as a general rule they either take the game too seriously, or opt for classes that play like their CoD-playstyles.
Do you know how frustrating it is when you're medic as offense and your entire team is engie and sniper? That's pretty much a summary of playing with the gibuses.
They also more often than not don't respect the peace heavy.
"sky is BUDDER KING OBEY JOIN US BUDDER BUDDREEEEEEERERERER" "pewds is genus u dome barel become a bro lolz" "o ur not BRONY well join the HERD JOIN JOIN JOIN" joinjoinjoinjoinjoinjoinjoinjoin Every fanbase has... THAT group of people. I don't mind if you show that you're a fan of something, as long as you don't slide it down my throat.
I then used my leet hacker skills to take my laptop to Geek Squad to remove the virus that somehow uploaded itself onto my computer.
After I came back from Best Buy, my other computer telepathically picked up my best friend and flung him against the wall, making him explode. At least the virus on my laptop was gone though, so all in all I think that was probably just about the best day of my life.
Somehow, when I clicked on this thread, I kind of expected it to involve MLP in some way.
Anyway, to go with the topic, here's my list of fanbases with a lot of annoying people. Note that I'm just saying there's a lot of annoying people in these fanbases, not trying to be offensive to people in those fandoms (remember, I'm a fan of many of the things listed).
Gaming in general (full of console wars, console vs. pc wars, hardcore vs. casual wars, fanboys who think their preferred console, genre, series, etc. is the best, and all the rest suck, people taking video games WAY too seriously, etc.)
Runescape (whines every time the game updates)
MLP: FiM (People overreacting about anything and everything, declaring the show RUINED FOREVER with every new episode, going out of their way to mention the show even when it's inappropriate to do so, so on, so on, so on. Seriously, I like the show and these people embarrass me so much I dropped the "brony" label.)
Apple (where do I even start?)
The Legend of Zelda (impossible to please, people whine about every new game no matter what)
Super Smash Bros. (People taking the game WAY too seriously)
EDIT: Also, any anti-fandom counts. They act like they're better than the crazy fans, but they're really just as annoying. I see them as the antithesis of crazy fans, actually; instead of being obsessed with the show/game/movie/etc., they're obsessed with hating it.