The plane will take off.
If it makes you feel any better, it took me moment to figure this out.
To simplify, let’s assume there is no wind and the plane’s wheels can turn on frictionless bearings. The pilot fires up the engines. Let’s figure out the net force on the plane. The engines are producing a force on the plane in the direction of travel. The plane is not yet moving, so there is no drag. Because there is a net force on the plane due to engine’s thrust, the plane accelerates in the direction of travel relative to the ground (and the air). As the plane starts moving relative to ground (and the air), the treadmill kicks up. However (and this is very important), because I have frictionless bearings, no force is exerted on the plane by the motion of the treadmill. The motion of the treadmill just makes the wheels spin twice as fast as they would normally.
The plane continues to accelerate relative to the ground and the air until it reaches takeoff speed, at which point it takes off. If takeoff speed is 100 mph, then the plane is moving relative to the ground (and the air) at 100 mph. At the same time, the treadmill is rolling backwards at 100 mph, so the relative speed difference between the plane and the treadmill surface at takeoff is 200 mph.
The real-world situation (with friction in the wheel bearings) simply means that the plane has to overcome a bit more friction to accelerate itself to takeoff speed.
All of you folks trying to figure out how this would work in the real world need to relax, IMHO. It’s a thought experiment, for crying out loud.
And all of you folks who think that the treadmill somehow is exerting a backwards force on the plane are apparently thinking that the plane’s brakes are on, or that the plane is bolted to the treadmill surface.