THe plane will take off. Period. End of discussion. The engines push the plane forward. When the treadmill tries to counteract the force of the engines, it has no way of "grabbing" the plane and pulling an equal and opposite force upon it, the wheels will simply spin. Since the treadmill can only cause the wheels to spin and not "pull" the plane backwards, the engines will continue to push the plane forward, creating lift over the wings (not the flaps, BTW, they create more drag on the plane when it is landing, you generally have your flaps in a neutral position when taking off and extended when landing) and causing the plane to lift.
There are 4 forces at work on an airplane at all times. An imbalance in these forces causes movement of the plane in the direction that is stronger than it's opposite force.
1. Thrust of an engine (pushing forward)
2. Drag of friction (usually of air, and in this case the treadmill)
3. Lift of the wings (lifting upward)
4. Pull of gravity (pulling towards the earth)
Looking at these four forces, the engines thrust forward, the treadmill tries to increase it's opposite pull, but it can't because the wheels will just spin and it can't transfer its energy to the plan itself. Thrust is the greater force and will propel the plane forward. As the plan moves forward, air begins to move over the wings and through the magic of the curved surface of the wings and Bernoulli's principle, lift greater than the pull of gravity is created, and the plane takes off.
The only way this plane would not take off is if the plane were on skis or the wheels were locked in place, or some other method of the treadmill to transfer its forces into the body of the plane, none of which is presented in the original question.
I won state level awards in school for my presentation/reports on airplane flight and balsa wood gliders I made and flew. Heck, I even got 9th at nationals one year. Just thought I'd throw some credentials in there.