You folks are silly.

The plane will not move (or it will if you add the element of time).

Break it down and simplify it, we just need to know if an object on the treadmill would move forward. If it can, then with wings, it can fly.
Let's remove the wings from the plane and the type of thrust from the plane, hell, do it in a vacuum. Now we have an object on a treadmill, it can be a box with 4 wheels, maybe a Hummer.

Apply a force to that box with 4 wheels, maybe with your hand, or maybe with a jet engine like in the scenario. If the treadmill matches the rotation of the wheels exactly, our object, box with no wings, Hummer, whatever, will remain in the same spot.

By matching the speed of the wheel rotation exactly, you have basically created 100% resistance, just like a wall. Park a plane against a wall, and see how far it gets.

Let's turn off the engine. Now, spin that treadmill up to 150 mph, roughly takeoff speed. The wheels on the plane are turning at 150mph as well. The plane is stationary.

Walk up a down escalator and match it's speed. Imagine that the escalator would speed up if you started running, keeping you in the same spot. Imagine a friend giving you a push from behind, making you run faster, but at the exact moment, the escalator speeds up again. Still can't get upstairs to Starbucks.

The only way it could actually move forward is from latency of the treadmill system adjusting to each increment in rotational speed and falling just a tiny bit behind. That was not part of the equation though.

PS: More pictures of Evangeline Lily please.