The Bills went to (and lost) 4 Super Bowls in the 90s with the no huddle offense.
The problem is it burns out your own defense too with significantly less time of possession.
The other problem is that it limits the versatility of your offense if that's what you do strictly. That's really bad for the playoffs, IMO, because all the other team has to do is watch the film and learn the small rolodex of plays that you use for the ENTIRE game.
I think having a no-huddle offense is a very nice ace up the sleeve, but it should be used as such; when it's needed (clock management, throwing off the defense's game plan). Being able to change the play at scrimmage is huge, too. This is why a player like Peyton Manning is inordinately effective and important. Or... was. This is also why the Packers are so hard to contain. They can beat you in so many ways.