It's amazing how obvious it is that some of you have never taken a basic physics course.
Ok, so how does a plane with its engines on not accelerate forward on a treadmill?
There are two things that can restrict it, one is dumping energy into the wheels of the plane, the other is dumping energy through internal friction of the wheels on the plane.
I'll give simple examples to explain both, and then you can just extrapolate them out to the plane and treadmill. But before I do that, I will say that in this scenario, it is a requirement that this treadmill is capable of insane accelerations, because when you work through the physics and calculate the accelerations needed to stop a plane with jet engines, the accelerations are very high. So, on to my examples:
Dumping energy into the wheels of the plane:
If you are standing on rollerblades on a treadmill and the treadmill starts, you initially have to hold onto the rails as the treadmill accelerates and your wheels accelerate to match the treadmill's speed. Why? Because your rollerblade wheels want to stay at rest, and you have to transfer energy into them to get them to spin. So you holding onto the rails as they accelerate is exactly equivalent to a force of a engine pushing your forward (holding you in place). Now the faster your treadmill accelerates, the more force you're going to have to apply to those rails to keep yourself from going off the back. So you can see that as acceleration rises, so does the force necessary to keep you on the treadmill. So, given that, and assuming a treadmill with very high acceleration capabilities, it could counteract the force of the jets trying to push the plane forward. But there's more...
Dumping energy into internal resistance of the wheels:
Now you are rolling on the treadmill, let's say at 10MPH on your rollerblades. If you let go of the treadmill, there is internal friction in your wheels, and they will begin to slow, and you will slide off the back of the treadmill. So there is a force required to keep you in place on the treadmill. This is an additional force, that in the problem, would be provided by the force of the airplane engines.
So the force of the engines can be counteracted by an acceleration of the plane's wheels, and through internal friction in the plane's wheels.
It is possible, assuming a treadmill capable of extremely high levels of acceleration.
And some of you are getting mixed up with crazy stuff. For example saying the treadmill matches the force of the engines, not the speed and such. If the plane is stable on the treadmill (not moving relative to the ground), then of course the treadmill is matching the force of the engines AND the speed of the wheels. If the forces are equal, there will be no linear acceleration down the treadmill, so the speed of the treadmill belt will exactly match the speed of the plane's wheels.
Digest that and let me know if you have any questions. I fully understand this, so saying "No! You dumbass!" will get you nowhere.
And I can make an equally compelling argument using physics for why the plane WILL take off, if we assume that "speed" means air speed.
Ok?