I've heard conflicting answers, and would like to see the record set straight:
An jet/propeller airplane is traveling on a giant treadmill at takeoff speed. Will the plane takeoff, or will it remain on the runway, and why?
An airplane's propulsion does not depend on friction between its wheels and the runway so the relative motion of the runway to the body of the airplane has no effect on the plane's motion$^1$. For example an airplane can take off from ice, where the friction between the wheels and the runway surface is effectively zero.
So the plane would take off normally if you placed it on a giant treadmill running at any speed.
An jet/propeller airplane is traveling on a giant treadmill at takeoff speed.
With respect to what does the plane have takeoff speed?
I believe the following two statements are uncontroversial:
(1) if the plane has takeoff airspeed (or greater), the plane can takeoff.
(2) if the plane does not have takeoff airspeed (or greater), the plane cannot takeoff.
Now, the plane's airspeed and the speed of the plane with respect to the treadmill belt are, within practical bounds, unrelated.
Thus, if you only specify the speed of the plane with respect to the treadmill belt, we don't know what the plane's airspeed is so we can only refer you to (1) and (2) above.
To be clear, it is possible for the plane to have both takeoff airspeed and, say, twice takeoff speed with respect to the treadmill belt.
For concreteness, assume the takeoff airspeed is 100 mph. Then, assuming low enough friction and strong enough wheels/tires, there is nothing to prevent the plane on the treadmill from having an airspeed of 100 mph and a speed relative to the treadmill belt of 200 mph.
The plane's propeller pulls the plane through the air as normal and the wheels rotate at twice the speed compared to an identical plane alongside on the runway next to the treadmill.