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A plane is standing on a runway that can move (like a giant conveyor belt). This conveyor has a control system that tracks the plane's speed and tunes the speed of the conveyor to be exactly the same (but in the opposite direction).

Will the plane be able to take off?

I wouldn't think so. You need speed to attain lift. How you gonna get that if you're stuck in place?

Here's an argument to that: "Oh, wait a minute. I was thinking of a car. Yeah the airplane will take off because the thrust isn't dependent on the ground, it's just that when it DOES take off the wheels will be spinning twice as fast."

turn the jets on full blast and lets see how fast that tread mill can go

It would work if the treadmill was replaced with a wind tunnel.

DT_Battlekruser is correct. It doesn't matter how fast the plane's wheels and the treadmill are going, if the air around the plane is stopped relative to the plane, it cannot take off because no lift is being generated.

Now instead of a plane, let's say a space shuttle right before lift off is put on this treadmill horizontally and has wheels on it. Would the shuttle be able to fly?

Right before lift-off, so the engines are gunning, yet it is horizontally placed. You've built a rocket car.
Run away from the site as fast as you can.

Treadmills are proven inefficient in aeroflights.

No, because the space shuttle is not built to either take off on its own or take off using its wings. However, unlike an airplane, a space shuttle and its boosters placed on a vertical treadmill would be able to take off, because it does not depend on airflow.

Exactly.

Instead of a treadmill, a propeller would be way more efficient.

Why do you asking this?

Assuming the treadmill has a coefficient of friction > 0, then at some point it would create enough force that it would stimulate movements in the air and the shuttle would lift off. (Assuming also that such a treadmill could withstand the forces involved with no maximum speed limit.)

Well, the question then is what aerodynamic environment you're running it in. In a wind tunnel it might get going fairly quickly, whereas sitting out in the middle of the ground it might just get too turbulent and never work properly.

Anyway, I think we're assuming the treadmill doesn't affect the air, only the wheels of the airplane/space shuttle/whatever.

Sure is! One of my favorites is where you have a large quantity of water floating in the air; if you swim around inside it you're okay, but as soon as you stick part of yourself out the bottom, you start falling, and the more is sticking out the faster you fall. Infinitely hard materials also provide for some interesting scenarios, such as FTL communication. And of course there's always the good old seeing yourself in non-euclidean space thing.

it would be a better reality to put that amount of water in space, and make it fall through the atmosphere. Walla, broiled humans.

Yes

Since there was no specification on what kind of plane it was, I assumed it to be a Harrier.

Well doesn't the conveyer belt have to have a coefficient of friction > 0 in the first place? Otherwise if it's at 0, then belt will be just like ice (actually, smoother than ice) and the plane's wheels will be moving but the plane itself wont move, even with the conveyer belt stationary.

Because if the substance actually did have a coefficient of friction = 0 it would mean that it is not made up of atoms, and is rather a mess of smaller particles, which would be so minuscule that the air in the room would be unable to differentiate the electronegativity on the surface, from one particle to the next. That is of course assuming that the belt is indeed a solid.

If, in the strange case that the treadmill was a liquid or gas, the fact that the tires themselves have volume, would create friction and drag in the treadmill. Of course this would mean that the plane itself would have to be more buoyant that whatever the treadmill is made of. So if we assume that the room pressure remains constant, while the treadmill (which is theoretically made of gas.) is made to have less and less drag, which would mean thinning out or depressurizing the gas in the belt; the plane would simply float upwards. This is not because of the treadmill, but simply because it would be less dense then the air in the room. You could then rightfully argue that the plane was a blimp.

lol

Actually, you're right, and I was thinking about that too. If the coefficient of friction was zero, the plane would remain in place and its wheels would not turn, no matter how fast you ran the conveyer belt. So for these purposes I think we're assuming either that one, the friction only affects the plane's wheels and not the air, or two, the belt cannot go nearly fast enough to get a high enough airflow without destroying the plane first.