What happens the transistors in a chip as the temperature increases?
The transistors don't just "fail" at some given temperature.  For silicon devices, as temperature increases, the devices get slower and their leakage increases.  At some point their performance drops off to the point that the device that they are part of, be it a gate, flip flop, ASIC, FPGA, or a microprocessor doesn't meet it's spec'd performance.  For industrial grade parts, this is usually 70 deg C.  For mil-aero parts, it's 125 deg C.

Also, other parts of the system will fail long before the silicon starts to melt.  Silicon has a melting point of approximately 1400 deg C.  The solder that attaches the part to the board has a melting point of around 185 deg C, depending on the type of solder.