Can an electric heat mat cause fake touches on a capacitive screen?
The problems you are describing are typical of EMI disturbance. Your tablet/phone with touch screen is a typical class B device which must accept any radio disturbance (up to a level) but may misbehave or stop functioning properly when a disturbance is applied. 

Your heat mat is most likely PWMed to regulate the temperature and if the implementation isn't good in this regard, the dV/dt and dI/dt flanks are high with the load acting like a large antenna, you have an EMI problem. 
EDIT: With your updated information that the problem still exists even when switched off but plugged in tells us that the power supply is the root cause of your problem. It will increase with load. You can try using clamp-on ferrites or of the wiring is detachable, get a toroid core and wind several turns around it to form a common mode filter. If the power supply is something simple, like a 12 V "dumb" supply and all control is done is the mat itself, you can try a different (approved) power supply. This will also increase you chances if suing the manufacturer who most likely bought and bundled a power supply with the mat and you would not believe how many no name Chinese manufacturers of non-EMI compliant manufacturers they have to choose between.


Your phone/tablet will be subjected to higher and higher field strength the closer to get to the mat and at some point something will misbehave. Exactly what will "give" first it difficult to say but something physically large, like the screen falls into the high probability category.