Can planes take off from a treadmill?
Everyone collectively went "Oh god, not this one" because this same question has sparked some intense debates in the past. Aircraft rely on airflow over the airfoil (wings/tail etc) to produce lift - which is independent of the movement of the tires. This means that with enough air going over the wing, the aircraft will fly even if it isn't moving forward at all relative to the ground.

This is why aircraft on ramps at airports need to be tethered to the ground. This isn't only to prevent them from rolling around, but from taking off should the air flow get fast enough over the wing.

If you are interested in an entertaining segment, the MythBusters did a fairly scientific experiment of this. 
                
             Although it isn't perfect, I think it does a good job of explaining the concepts.