The auction house has absolutely no effect on drop rates. There are conspiracy theories and misunderstandings but I do want to re-iterate, the is NO interaction whatsoever. Bashiok mentioned earlier that we took the AH into account, so let me expand a little bit on that.
The drop rates were tuned for a player who would never use the Auction House. For the majority of internal development we didn't have an Auction House, we all played using our own drops only. I've personally leveled multiple characters from 1 to 60 internally before the game came out using only drops that I found - we all did.
When we say we "took the AH into account" that means it's one of many factors. ie. some players will choose to play without trading, some players would play in a group of 4 where they share drops among each other, and some (as it turns out, many) players would use the AH.
Three weeks after launch player's gear is much higher than what we were expecting. When I killed the Butcher on Inferno for the first time I was using a weapon with 492 DPS. There are also certain passives which are much more powerful than they were during internal development. One With Everything, for example, was basically never used internally because we didn't have an auction House. With the auction house, it feels like a mandatory passive. In retrospect we should have seen it coming. In the game's current state though, it's a powerful Monk ability that gives Monks a big survivability boost and has some interesting (some would argue fun, others would argue negative) effects on gearing.
I consider playing without the Auction House to be a very fun way to play the game. I'm personally planning on rolling some new characters that I'll set aside to be "no-AH/no-twink" characters. Much like in D2 when I would make a new character with a friend and we'd agree with each other not to twink our characters out.