DO: be a good neighbor. It benefits you AND is the right thing to do! If you see someone struggling in a fight, help them out, even if it just means tossing a buff or a heal. Sure, you won't get any loot or XP out of it, but you'll feel better about yourself and you'll benefit another person. Oftentimes, if that person has buffs that work on you, they'll cast it on you as thanks. (Note: For this reason, always help out druids. Also note: always buff people who help you out. It's part of being a good neighbor.)
And remember: what goes around comes around. Sure, that particular person may never help you out himself. But if you find yourself struggling and another player passes by, would you want to be in a culture of people who help each other, or a culture of it's-none-of-my-business? Creating culture requires YOUR effort, too. So be a good neighbor. (This is doubly true for the PvP servers.)
DON'T: Be a jerk. I really don't have to explain this, because everyone knows what it means. Yet I have to mention it because some people inevitably act like this. Ninja-looting, spamming the chat forums, pestering or following people for no good reason, letting people die so you can kill the mob they were fighting, all are examples of being a jerk. (Also: some of the more egregious PvP actions such as corpse-camping and repeated ganking.) Being a jerk angers other people and there's no real benefit to you, PLUS you'll feel guilty about it. It hurts you in the long run, in the form of blacklists and a hostile general community. And don't go trying to say that you're role-playing a character that's a jerk, because that's just stupid and no one really buys it. So don't be a jerk.
DO: Learn how to be a good group-mate. Know how your character works and what it's capable of. For example, if you're a secondary healer, it becomes your job to watch health bars, too. If your character can tank but is not the primary, the group is counting on you to handle adds. Do your best to help out the group as a whole. It's tempting, particularly with some classes, to forget everything and focus on blowing the monsters to kingdom come. But your healer won't be thanking you when you draw the aggro by overnuking, and he has to heal you a lot more which means now the aggro's on him, and suddenly the group wipes. Or the healer goes OOM and needs you to cover while he regens, except you blew your mana for burst DPS, and the tank dies and you wipe.
Being a good group-mate is about communication, also, especially when one or two of the party members is very experienced. Be prepared to take advice, tell your party members IN ADVANCE if you need a break (they might need one, too, but will resent you if you leave them hanging), and keep everyone alert about adds or mana situations.
DON'T: be mean to the healer. While being mean to your group-mates is a bad general policy, it especially hurts when you're mean to a healer. The healer is the only person who doesn't get to really fight, he's got the most tedious and thankless job in the party, and he's always the first to be blamed when something goes wrong. Priests, despite being fairly strong as a class, are the least-played class in the game for just these reasons. It's one thing to say "Why didn't you heal me" to a secondary healer, as that person might not completely understand his job. But NEVER say "why didn't you heal me" to a primary healer, because he is VERY familiar with his job. There are a few bad healers out there, granted, but the rest are people who are really nice guys and who really, really want to help. Be nice to them. Thank them for a job well done-- after all, no healer=dead party, so by definition any success is because the healer was on the ball. It's all they really ask for, so don't be shy; just make a habit of thanking the healer once or twice a run. In fact, I think it's so important it'll be it's own topic.
DO: Be nice to the healer. See above.