There are some good points being raised. I was assuming you'd be flying through a small tornado (F0-F2), a larger tornado like an F5 would swallow anything alive. I don't think planespotta's example is entirely relevant because thunderheads are large, ugly clouds that you can easily spend over an hour in. If you flew through a tornado, your plane should be in and out within a matter of seconds. And I like photon's theory of momentum. I agree that if a large plane is traveling fast enough, it's momentum would be much too large to stop.

My guess is that a large, airplane with enough speed could make it through. That's not to say the airplane wouldn't be damaged severly, it would probably have to make one quick emergency landing afterwards.