Here's my take on it

In the theoretical description of the question, the end result will be a plane throw off the end of the conveyor-belt-runway and destroyed.

In a real-life attempt at the question, the plane will take off.

Explanation:
If the conveyor belt actually move at a rate exactly the same as the relative speed of the plane on the ground (which at ground level would be the same as its airspeed, as soon as there is enough thrust to move the plane forward, the speed of the conveyor belt (and the rotation of the planes tires) would increase exponentially towards infinity to compensate. This speed would result result in catastrophic failure of the plane's wheel bearings and axles, and the plane would then be violently tossed off the end of the moving runway by the infinitely-fast-moving conveyor belt since there was no longer a spinning wheel to aid in friction loss between the plane and runway.

In real-life, it would not be possible to create a conveyor-belt-runway that could increase in speed to compensate for the thrust of the plane's engines, so the conveyor belt would never get up enough speed to keep the plane in one position, and the plane would takeoff.