Most of my alts are healers and tanks - but my dps alts who are often low level do pull ahead of slow tanks - but only if I can solo the upcoming pull (which is possible for pretty much any geared class pre-50, and many classes well after that). My tank alts just pick up any mobs other people pull though, and my healers don't let people die no matter why they pulled (accidental or otherwise) unless they're also being a massive douche in chat - but pulling extra mobs in dungeons? Oh noz.
Think of it as a challenge to your ability, hopefully you aren't challenged by a tank slowly pulling and tanking one group at a time - and if you aren't challenged - what's the point?
I don't mind if someone else ends up pulling because of CC reasons or perhaps to do a better LoS pull. If I am in a pug and the tank doesn't seem to mind the other person pulling first (usually NOT the case, but it does rarely happen) then I just deal with it. Of course if they end up dying because they take too much damage or the group complains I have to drink more often than normal because I'm spam healing, then I'll inform them of the easiest solution to that problem. I haven't had a ton of trouble with it but the issue has occurred a few times since I've been playing. If I'm not busting my back keeping them alive, I just deal with it. But if it's causing added stress because dps is pulling and the tank isn't as geared, then we have problems.
I think it also depends on if the tank is bad or not. If the tank sucks and a dps i pulling because of that, heal the dps. If the tank is slow as hell, let the dps pull and heal them, if its pre-80 dungeon let the dps pull and who cares who tanks. Hell i've tanked and healed on my priest in pre-cata dungeons.