Why exactly do chips start malfunctioning once they overheat?
To expand on other answers.


Higher leakage currents: this can lead to more heating issues and can easily result in thermal runaway.
Signal to noise ration will decrease as thermal noise increases: This can result in a higher bit error rate, this will cause a program to be misread and commands to be misinterpreted. This can cause "random" operation.
Dopants become more mobile with heat. When you have a fully overheated chip the transistor can cease being transistors.This is irreversible.
Uneven heating can make the crystalline structure of Si break down. A normal person can experience by putting glass through temperature shock. It will shatter, a bit extreme, but it illustrates the point. This is irreversible.
ROM memories that depend on a charged isolated plate will be able to lose memory as temperature increases. The thermal energy, if high enough, can allow electronics to escape the charged conductor. This can corrupt program memory. This regularly happens to me during soldering of ICs that are already programmed when someone overheats the chip.
Loss of transistor control: With enough thermal energy your electrons can jump the bandgap. A semiconductor is a material that has a small bandgap so that it is easily bridged with dopants but large enough that the required operating temperature does not turn it into a conductor where the gap is smaller then the thermal energy of the material. This is an oversimplification and is the basis of another post, but I wanted to add it and put it in my own words.


There are more reasons, but these make an important few.