Nope.

Stand on the ground, holding a wheelbarrow, and put the front wheel on a treadmill. Speed up the treadmill, and observe how the wheel spins at the same speed as the treadmill, but the wheelbarrow stays in one place.

Then see that if you push the wheelbarrow forward (on the moving treadmill), it moves forward regardless of how fast the treadmill is going. Heck, make the treadmill go in reverse, and observe how you can still push the wheelbarrow forward and back.

The original question is flawed in assuming the treadmill can hold the plane back. It can't.

HOWEVER, if the wheels of the plane were driven by the engine to be the same speed as the treadmill, then yes, it would never have any forward motion or airspeed, and never take off. But I've never seen a plane with powered wheels.