How can Munchkin Quest be played quicker?
I've had this problem in the past as well, and the best solution that me and my friends have come up with is to play the game less like the rules say, and more like Betrayal At the House on the Hill.

Which is to say, when moving, you only get to explore until you have an encounter.  This could be fighting a monster or a trap or some such, but once you've had to resolve anything that isn't just drawing a card, you stop and the next person's turn begins. The downside is that your dungeon doesn't grow as fast, but the upside is you don't end up with each person having to go through the combat scenario 3 times each turn, and you'd be amazed how much that speeds things up (because you don't have as many people trying to play modifier cards).