The topic title says it all: which quests do you refuse to accept? OF course it's more fun when you explain why you hate it so much.
I personally never take quests that involve a lot of travelling. Even though I am a mage, those quests are usually not worth the trouble. Going from Theramore to Feralas without a flightpoint (the only good thing is that you'll get that along the way) to kill 3 creatures... in the same time you do that your friend can boost you an entire level up through Zul'Farak!
I do take messenger-quests sometimes, though it can be tiring, especially in Darkshire where you are running between 3 npc's and basically doing nothing. Free XP though, so no real complaints.
I'm usually stubborn on the low-drop quests. I keep on grinding until I have the item. The fact that my boyfriend keeps telling me it's silly probably makes me even more stubborn!
With the animal-killing, I got especially annoyed when I was a young and unexperienced mage with mana-issues. I didn't feel much better when I levelled my priest, because the priest couldn't conjure water or evocate. Only after I got a hunter and a rogue I learned to appreciate these kind of quests.
I don't have a problem with group-quests, but they actually have to add something. Ring of Blood and Amphitheatre of Anguish are groupquests I really enjoy.
Obviously, there's always a Donald that accepts the next one before the rest turned in their quest, but luckily the Elites go down fast and grand a lot of XP.