I can not tell you what happens if there is a positive value for the temperature above 20000ft, but here is some general information:
For this issue I will be talking about Airbus planes, as they are known to be very "smart" planes. There are hundreds of algorithms that try to prevent wrong data input from being an issue. But those do not always work.
Take Qantas Flight 72 for example: According to Wikipedia the airplane all of the sudden pitched down violently, as if it was trying to prevent a stall from happening. This was because the average of the two angle of attack sensors was way out the box. Though there was a (very smart) algorithm rechecking the values, it could not deal with multiple wrong values with a certain amount of time - and made the airplane pitch down.
This being an example where the plane actually acted by itself due to a faulty value, here is the general case: If the airplane notices odd values, which it will usually recognize (e.g. a positive temperature), it blanks out the display so that a pilot can not be mislead.
When flying smaller airplanes, you often practice "partial panel flying", where your flight instructor covers up some instruments and you have to fly the aircraft with alternative instruments. As you can see this is well practiced and
shouldn't be a problem. Unfortunately though, this makes flying a lot harder which can cause crashes to happen (AF447,QZ8501, and more).
Furthermore, whenever the autopilot finds itself in a situation that it can't handle, for example when the pitot static system fails, it will disconnect and refuse to connect, giving the pilot full control (or rather as full control as possible on an Airbus) of the aircraft. This disengagement ensures the plane not to do something stupid due to wrong inputs.
Conclusion: For redundancy most systems exist two or three times working parallel to provide backup, same goes for sensors. If all data inputs report something obviously wrong, the pilot will not see this wrong information so he can concentrate on what he has.
The answers to this question can also give you a little more information about computer redundancy on aircraft