Argh, all these airplane/treadmill threads are so incredibly frustrating. The plane WILL accellerate as normal, (perhaps ever-so-slightly slower due to the increased friction from the treadmill) and take off. The wheels will be spinning twice as fast as normal, but this is irrelevent as the engine is delivering power to a propeller, which is dragging its way through stationary air.
As for a helicopter on a turntable, if you ignore the effects of the tail rotor trying to keep the heli from madly spinning, and also ignore the fact that spinning the entire helicopter at high speeds will probably throw the whole thing apart, no, it won't take off. (if the turntable is going the opposite direction to the rotor at the same RPM as the rotor) The reason being that the engine is trying to rotate the rotor at X rpm, and if you spin the heli in the opposite direction at the same speed, you end up with a stationary rotor blade and a very very rapidly spinning helicopter.
To make things even more confusing, if you put wheels on the helicopter in such a way as it could turn or pivot around it's axis, and factor in the tail rotor again, the turntable will spin but the heli will stay facing the same way (the tail rotor will counteract both the force of the main rotor trying to turn the heli, and the added friction of the wheels rolling on the turntable) Therefore, you end up with a rapidly spinning turntable, a stationary heli that is in contact with the table with some sort of circular array of wheels, and the main rotor will rotate as normal, allowing it to take off.