What happens if a plane flies too high?

Depends.

1) the Engine might struggle in the thin and Cold air, stopping the plane from getting any higher.

2) the pilot might struggle in the thin and Cold air. Might faint from lack of oxygen. The plane might go into a shallow dive, allowing the pilot to recover and regain Control. Or might tumble uncontrollably and crash.

3) The plane might be able to get so high that the Control Surfaces are no longer effective. If that happens, it might enter thicker atmosphere at an angle incompatible with controlled flight and go into a catastrophic dive.