Why higher temperatures shorten the lifespan of a PC component?
Melting and burning are just extreme extensions of the process.
Generally, changes happen faster with increasing temperature and material properties generally alter. As devices are usually designed to perform best "as supplied" changes are usually bad. For example, a lubricant will be designed to work best at the temperatures expected to be encountered. Increase the temperature above the expected range or above the range at which it was feasible to optimise performance and eg lower viscosity and increased oxidation rate will decrease lifetime. (Lowering temperature markedly may also adversely affect things like viscosity but generally for different reasons.)


Chemical reaction rates increase with temperature, including oxidation from atmospheric Oxygen, impurity reactions etc.
Plastics "soften". Materials' mechanical properties generally degrade as -they approach melting point or glass transition temperatures or ... 
Diffusion rates increase. Both at eg metal-plastic sealing boundaries and in materials that rely on differences at boundaries.
Components that rely on internal liquid content dry out. Notable examples are electrolytic capacitors and batteries.
Lubricant properties change (eg motor bearings). 
Defects in materials are liable to form faster and propagate more rapidly and further. (This is a major factor in eg long term LED lifetimes). 


In some cases, attempts to address temperature issues can be counterproductive. eg how long will a fan last for a given bearing technology and cost etc. 

"Collateral damage" can occur eg aluminum electrolytic capacitors are sensitive to heat degradation, largely through electrolyte dry out. Placing them near a hot device may cause early capacitor failure even though this could not be predicted from analysis of the circuit.

Some products are commonly used without regard to their time &amp; temperature susceptibility. "Hot melt" adhesives are often used for component fastening but will lose their grip on many surfaces in weeks to months. 

Temperature effects can be regenerative (positive feedback). eg Iron powder cored toroids can provide superior performance for power conversion inductors in the 10's of kHz to low MHz range. The cores are formed with an organic binder which is temperature affected, giving it a design lifetime. As the binder ages core losses increase leading to increased temperature rise leading to more rapid degradation leading to ....  An iron powder core handling significant power can go from nearly 100% OK to a smoking ruin in a very short period relative to total design life as it hits the end of its core design life. Similarly, electrolytic capacitors handling high ripple currents will have increasing ESR as electrolyte dryout occurs, leading to increased losses and increased temperatures leading to ... .