Since Quest Rogue seems to be my mainstay just now, I can offer a few things you can try. Yes, Dirty Rat can mess you up fairly quickly as it doesn't allow Battlecries to trigger. Remember, for any quest deck the major drawback is you are giving up one card in the beginning to start the quest. So first turn you've only got 3-4 other cards to use and usually only 2 of them have any early game value. If someone plays a dirty rat and pulls a Ferryman or a Brewer I needed to claw back a fourth elemental later on, you just cost me a turn or three to get to my Crystal Core. That is catastrophic to Quest Rogue's tempo.
The other effective tactics I've seen are either to aggro so fast on the face that my 5/5s arrive just as I hit 0 HP, or to have some super effective board clear. The good old fashioned Pally changing all health to 1 and then either consecrating the field or having the spellcaster do it comes to mind.
Also, if they have committed to playing and withdrawing a certain card (like swashbuckler) and leave it on the board between turns, do anything you can to kill it before the next turn. They might get it out 3 times very fast, but if they lose it the 2nd one might be 10 cards away in the deck (or they might have it in hand but what can you do?) And it is hard to start pushing a second card type to four once you've started spending cards pulling something else back.
If a Quest Rogue has their early cards, draws decently, and doesn't have their tempo squashed they are almost unbeatable IMHO. So screw up their rhythm.