I thought about this all day and told my dad and some people at work. They all initially thought the plane wouldn't fly and we argued all day. In the end I won them over to believing that it would fly.

Let's look at the original problem again. I'll cut and paste it here...

"A plane is standing on runway that can move (some sort of band conveyer). The plane moves in one direction, while the conveyer moves in the opposite direction. This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same (but in opposite direction).

The question is:

Will the plane take off or not? Will it be able to run up and take off?"

We can't reasonably assume much more than this stuff....
It's a plane that is capable of flight, meaning it has enough trust to go forward really fast using AIR as propellant like most planes do. We have to assume it's a plane that won't have powered wheels because most don't. We can't assume the air is blowing in any direction. In Physics (at least basic physics) we always assume "everything else is being equal" so we must assume that here to. Often when solving problems Physicits ignore friction or drag until later in the problem so we will have to ignore those too. Like friction in wheel bearings or rolling resistance. We can't measure that even though we all know it's there to some degree. We have to assume that the tires will handle any speed of rotation like land speed record cars that use solid aluminum wheels or whatever they use.

"Every action has an opposite and equal reaction"

Think about a guy shooting a shotgun. The powder ignites and explodes in all directions. The gun's barrel channels that energy to push the slug or shot out one end and the gun goes the other direction (into your shoulder) with an equal amount of force to the gun and the shot.

A plane works the same way. The props or jets suck in air and thrust it out the other end. That's how they power forward regardless of the surface they are on.

OK, Now get a clean sheet of paper for your brain, and think about a plane hanging from a pulley on an old style clothes line with pulleys on each end. The clothes line advancing towards the plane in much the same way as the conveyor in our problem. When the planes motors turn on, isn't it easy to imagine that the plane could travel down the line faster than the line could advance toward the plane and the plane could get lift under it's wings?

New clean sheet of paper for your mind. How do rockets work? They have a propelant that is burned and the gasses are THRUST out the engine end of the rocket. Remembering that "Every action has an opposite and equal reaction". The rocket is propeled forward with as much force as the burning and expanding gasses are going out the engine end. With me? Now lay that rocket down on our moving runway and fire it up. What happend? The rocket didn't care about the runway. Now slap some wheels and wings on it and fire it up again on the runway. What happened? The rocket plane didn't care about whatever the runway was doing! Now call our new rocket plane a plane and change over to regular props or jets. Do you now see that the plane will fly and that the runway has no bearing on that.

The original problem was written to make you think about the runway and be overly concerned about it.

I didn't spell check this post either.