The plane takes off in approximately the same distance necessary on a conveyor belt as it does on a regular runway.

Since Blue-Sky is mentioning his 20 years "aviation" experience, I guess I'll throw mine into the hat, even though it too has no bearing on the problem... I'm an airport designer. Civil Engineer. I design Runways, Taxiways, Aprons, and all things associated except the electrical & navigational equipment. I've been doing this for 6 years, now.

I assure you, the friction of the wheel bearings does not matter to the plane taking off. How's that? Because different surfaces have different coefficients of friction already. Asphalt runways have different coeff. than concrete. Significantly different on paper; absolutely no difference in the real world. Why's that? Because both numbers are so small in relationship to the take-off thrust of any aircraft that it doesn't matter.

READ BELOW COMPLETELY FOR AS SIMPLE AN EXPLANATION AS YOU CAN GET ON WHY THE AIRPLANE WILL TAKE OFF

An aircraft's engine PUSHES (jet) / PULLS (prop) the aircraft along the ground. This results in a horizontal force on the axles of the aircarft because they are fixed rigidly to the aircraft. The by-product of this force is it causes ROTATION of the tires around the axles. The tires do nothing more than rotate around the axle; they do not cause any horizontal movement.

If a conveyor belt were under the tires, it too would cause rotation of the tires around the axles. But, it can not cause any movement and/or restriction of movement of the axles, as the tires do nothing but free-spin around them. Again, the conveyor belt spins the tires, but does not exert any horizontal force on the axle. You can test this "theory" by jacking up your truck (get a whole axle up in the air), and spinning a free-wheeled tire from below. The tire will spin, but the truck will NOT move. This is not because you aren't spinning the tire fast enough; spin it as fast as you want, and the truck still will not move. The truck can't move, because all the force you apply goes into spinning the tire; none of it is a horizontal force on the axle.

Summing the forces in the horizontal direction gives you a positive force from the engines, and 0 negative force, as the conveyor belt can not move the axles of the plane regardless of how fast it spins.

From Newton, Force = Mass x Acceleration. So divide the sum of the Forces (the engine force only as shown in the above paragraph) and divide by the Mass of the plane. What you get is the Acceleration of the aircraft. The friction is negligable, as it is an extremely small number compared to the Force.

The net result is the aircraft accelerates down the runway, and eventually obtains flight once there's enough lift provided by the air passing over the wings.

END EXPLANATION

And, it will reach its takeoff speed at the same distance down the runway as it would without the conveyor belt. The additional friction of the tires as they spin twice as fast is not enough to make a significant difference. Dynamic friction levels off at a certain point, and for all practical purposes remains a constant level whether the tire spins at 150 mph or 300 mph.

Editorial Comments...:
While this is a physics, it's also a Statics & Dynamics problem. A problem like this would be taught in a 200 level Engineering Dynamics course, and any Engineer that can't answer it with only a fleeting glance ought to take some continuing education classes in a hurry...