Heat is produced by all the active circuits within a PSU.
The main sources are the switching transistors and rectifiers. Other significant sources include the inductors (resistive and core losses) and capacitors (ripple current losses).
High temperatures may damage any component - often the transistors will go first, as they get hottest. When a transistor fails, high voltages may be directed onto low-voltage circuits, causing catastrophic failure of capacitors and other components.
In the longer-term, capacitors are the most affected by high temperatures. The hotter a cap gets, the less time it lasts. The most likely effect of this is that its energy storage capacity gradually fades, and the voltage regulation ability of the PSU gradually deteriorates.
Good PSUs will have an 'overvoltage crowbar' circuit connected to the ouput. This circuit is seperate to the normal monitoring circuits, and is a fail-safe, preventing damaging voltages reaching the output. If the PSU malfunctions and produces a high voltage, the crowbar will kick in, and short-out the PSU; the PSU may shut down or be destroyed by the protection circuit.