The behaviors I find annoying:

The tank-wannabe:
DPS who want to be a tank, run ahead of the tank, stand on the far side of mobs when attacking, and pull the next group of mobs before the tank. Kind of annoying as a healer. Why can't the DPS just let the tank do their job? It's especially annoying when the tank is by no means moving slow.

Tanks desperate for heals:
I've had a few tanks who begin to run towards me or around me when their health gets below 50%. Don't worry. I got you. Running towards me or around me isn't going to make me heal you any faster. I won't let you die. Don't bring the mobs to me and the ranged DPS. Just focus on the mobs and I'll focus on you.

The chatter:
This one bothers me more than anything. People will suddenly stop moving or be stuck on auto-attack because they are likely talking to their FC or Linkshell. When their target mob dies they do nothing. Just standing their with their weapon drawn.
And what's worse, is the person who wants to make idle chit-chat in the dungeon.
When people see I play with English client, sometimes I get "Oh, you speak English?" "Where are you from?" "Can you speak to me in English?"
Do you really think the middle of the fight is the best time for this?

The stoneskin healer:
Because I mostly play healer, I don't play with too many other healers. But every now and then I get one who feels they must make sure every party member has stoneskin on before every encounter. I get it at the beginning maybe and before bosses, but it's not really necessary. Just takes too much time unless it's Stoneskin II.

The stoneskin paladin:
This Paladin will wait for you to use stoneskin before every encounter. Just standing around. But they won't actually ask for Stoneskin. So when I don't cast it, they cast it on themselves. Before. Every. Encounter.