NOTE: I'm assuming you have tuned your deck and all of the cards in it for the purpose of completing the quests.
The warrior quest can take a long time. Sometimes going into turn 9+ depending on what kind of removal you need. Playing against a high aggro deck such as the rogue quest you may want to hard mulligan for weapons or better stat minions to clear their early minions so they can't bounce them.
The priest quest is built for the long game. You want to play tons of minions with deathrattles and the only good early ones are the 1/1 elemental or the 2/2 mistress. You can mulligan your quest if you have no decent early deathrattles and you need to fight for early board position. You can also get some really good deathrattle minions late game and complete your quest for an easy win if you can get board control against a control deck.
DRUID You can safely do this against aggro decks for cheap removal spells in the early game. You need bigger minions with 5+ attack which usually come around turn 3 at the EARLIEST. If you need early game board presence I suggest doing it so you don't get too far behind.
MAGE You can if you are going for exodia mage. Mage quest takes a while and the only good 1 drops in most mage decks is mana wyrm if you're going for quest completion. You'll need tempo and you don't want to cast primordial on turn 2 and waste your turn if you're facing aggro.
PALADIN You don't. You need small minions and start casting spells to buff them asap to get board presence and spell bounce back.
HUNTER You don't. Works well in early game with 1 drop minions. No point in wasting the early game.
SHAMAN You don't. Murlocks are cheap and can go out in turn 1 with coin.
WARLOCK You don't. You want to be able to start discarding from the start of the game with your discard/draw mechanics.
ROGUE You don't. For the love of god don't mulligan your quest.