It can't fly, because it can't move forward. As soon as the plane begins to accelerate, the treadmill will instantly accelerate to a nearly infinite speed. Since rolling resistance will keep increasing as the treadmill increases in speed, the plane simply can't get moving through the air because the rolling resistance will increase to exactly counteract the engine thrust. If the rolling resistance isn't strong enough, the plane will start to creep forward, causing the treadmill to speed up, which will increase the rolling resistance which will halt the plane again.
There is only one exception, and it is the same thing that makes modern hard drives work... A moving surface will create a little bit of wind over that surface.
http://www.pcguide.com/ref/hdd/op/heads/op_Height.htmSo the treadmill itself will create a little layer of moving air along the surface. This is how computer hard drive read/write heads "fly" on the very thin layer of air on the top of the spinning hard drive discs, and why it is a really bad idea to try to completely seal a computer hard drive or run it in a vacuum. Assuming that the wheels don't explode from the nearly infinite rotation speed or the bearings don't give out, at some speed (very very fast), the treadmill itself would probably create enough wind to lift the plane up off of the treadmill, at which point the rolling resistance on the landing gear would drop to zero, and the plane would begin to move forward. It's a bit like if a stationary plane was hit with a gust of wind... It would essentially leap up and backwards relative to the ground unless the engine was strong enough to accelerate it forward through the air and assuming the wind didn't stop.
But you have to assume a whole lot of things that simply can't happen, primarily that the landing gear, wheels, bearings, etc. would survive the treadmill's acceleration to a near-infinite speed, as required by how the original question is stated. That simply can't ever be tested in full scale.
It would be possible to test a subscale model at lower speeds though... Just slowly increase both the engine thrust and treadmill speed so that the drag created by the rolling resistance always equals the thrust of the engine. At some point, the rolling resistance will be as great as the engine's maximum thrust, and there will be an equilibrium point where the plane doesn't move because the thrust of the engine at max power is exactly counteracted by the wheel drag from the treadmill. In a full-scale test, I'm not sure the rolling resistance would ever be enough without the treadmill either moving far faster than physically possible using known materials, or if the wind effect caused by the treadmill itself would create enough airflow over the wings to allow the plane to lift off of the treadmill.
I used to know what that wind effect was called, but I can't remember it and I'm apparently not smart enough to google it correctly either.
