The wheels are spinning passively, they have no role in moving the plane forward. Completely free-spinning except for an almost trivial drag. In any scenario except the magic exploding feedback loop, the plane moves as normal, but the wheel rotates twice as fast as normal, getting torque from the usual forward motion of the plane plus an additional torque from the equal motion of the conveyor belt. If you just think “wheels totally passive” you’ll come round to it shortly.
No I haven’t. The OP says that the conveyor spins at the same rate as the wheels. So when did the wheels start spinning? The only way that could ever be possible is if the conveyor belt had zero inertia (i.e. is effectively weightless) and yet had 100% friction against the wheels of the plane. The wheels on the other hand do have friction against the axis they are attached to and inertia to be overcome to start them spinning–which the conveyor belt cannot do.
If the magic conveyor belt can stay perfectly in synch without any delay (i.e. no difference in speed at the point of contact) how can the wheels ever spin? In order for force to be exchanged from one to the other, one of the two will have to tug at the other–but how can they “tug” if they are both moving at the same exact speed at the point of contact?
The OP says the conveyer belt matches the speed of the wheels, but moves **in the opposite direction **. That’s open to interpretation, sure, but a zero-rotation wheel on a forward-moving treadmill seems to violate the spirit of the problem.
And I should point out that automobile tires (well, tires in general) and the surface of the road move at the same exact speed at the point of contact–namely, zero speed. I think you’ll agree that a force is transferred between the car tire and the road in this case.
It can move in the opposite direction all it wants, but the opposite of zero is zero.
With 100% traction against the Earth, a moving car would spin the earth backwards slightly to accelerate itself without there ever being a difference in speed at the point of contact. So yes, my wording was incorrect for the problem. But, if you stand on Mars and push the car from behind with your finger, the friction of the axle will cause the Earth to rotate forward with the car.
The obvious answer to this is that it’s possible only if one uses scalar weapons! 
So you’re proposing that neither the wheels nor the plane nor the treadmill would move at all? In that case, what resists the thrust force from the engine?
Air.
The plane would move. And the treadmill to some extent or another would move forward with it due to the friction of the axle.
You are correct, without magic, they wouldn’t. In the literal wording of the scenario, the wheels don’t pull on the treadmill. The engines add a bit of thrust, the plane begins to roll forward, and the magic feedback mechanism signals the treadmill to move retrograde at the same speed. If friction exists, the wheels turn faster, further signalling the magic regulator to increase speed. In effect the system is a positive feedback loop that will cause the spin of wheel/treadmill system almost instantly to accelerate to Ludicrous Speed, in perfect sync. And the airplane will be moving forward during the few microseconds it takes for the system to accelerate to the point of either physical disintegration or relativistic effects (the latter being a can of worms I’m not qualified to open, nor will I try).
Of course if you had the sense of the conveyor belt reversed, which I’m thinking you might, then the wheels would not rotate at all and the aircraft would be catapulted forward as if from an aircraft carrier. I think.
Which would violate the spirit of the problem, since the OP specifically said the treadmill moves backwards.
Why does the plane “roll” forward? What is causing the wheels to turn?
It’s exactly like real-world physics, the wheels turn when the plane thrusts forward because they are reasonably free-turning and are in contact with a tractionable surface. The only difference in the magic scenario is that the surface is signalled to move backward at the same speed, which accelerates the wheels, in turn feeding back to the surface controller.
Backwards at the same speed as the wheels. If the wheels do not turn, the conveyor belt does not turn. But, the conveyor does have to move with the plane, which is separate from the wheels.
The only other posible solution, going back to the idea of the car sitting on Earth and you standing on Mars is that the Earth rotates backwards soooooo fast that as you push on the car from behind, the friction of the wheels and axle grows so great as to counterbalance your force. Yes, in this case the plane will not move. If you remove the friction of the axle as everyone here is trying to do, it becomes an impossible equation; the instant any force is exerted on the plane in any direction, the universe would implode.
The thrust from the engines pushes the entire aircraft forward through the air, regardless of whether the wheels spin clockwise, anti-clockwise, or not at all. When the air over the wings reaches the appropriate speed, the aircraft lifts upward. QED
But if the plane moved, then the conveyor belt did not work to keep it in place.
And if the plane didn’t move, then the wheels have no reason to roll–so the conveyor belt doesn’t move. (Gah, I’m doing so awful at getting all the thoughts out in one shot.)
If the wheels are able to turn, the plane is going to move forward under the thrust of the engines, no matter what the treadmill does.
Not if you add the friction between the axle and the wheels…
So yes there is a solution that behaves similar to the OP’s where the plane will stay still. But I don’t feel that the OP intended for there to be friction between the wheels and the plane. The only solution which remains is an inertialess conveyor belt that is rooted to the tires and if the tires don’t turn, it has to stay with the wheels. If neither the wheels nor the conveyor belt has inertia and there is no friction between the plane and the wheels, and then the plane is pushed forward, the plane will move forward and the conveyor will be pulled forward with it.
Hence
Say the aircraft needs to have an airspeed of 200mph to achieve lift, and this aircraft is on this treadmill and the treadmill is actively trying to roll backwards at the same rate the aircraft is rolling forwards. That just means that, at liftoff, the wheels are spinning as if the aircraft were going 400 mph. No big deal to the axles.