This is from one of pm's posts. Pretty much says everything you want to know:
Increased voltage leads to three reliability concerns directly and one problem indirectly. Apologies in advance if I have resorted too much to engineering-ese in this post... but it is a complex topic.
The primary reliability concern for high voltages is gate dielectric breakdown. At a high enough voltage the dielectric of the gate of a MOSFET transistor will 'breakdown' essentially like air breaks down in the presence of a high voltage differential between the earth and a stormcloud. In the case of a MOSFET, however, the breakdown leads to a permanent highly resistive path through the FET and the transistor breaking. So, high enough voltages will lead to a dead chip once the breakdown voltage of the weakest FET on the chip is reached.
Two other related concerns are PMOS BTI and NMOS Hot-E. Both are fairly complex topics - in fact I'm still not absolutely certain that researchers entirely understand the causes of PMOS BTI - so I'll gloss over the details as to specifically what happens. From a macro point of view what happens is that higher voltages on the gate lead to increased threshold voltages on the transistor over time. This leads to reduced drive strength (AKA higher channel resistance during "on"), and eventually the reduced current drive will led to a critical path failure.
Lastly electromigration - the physical transport of atoms under constant current conditions - leads to movement of atoms in the wires which eventually thins out the wires so that they become too resistive to carry current (leading to IR voltage drop on the power rails typically, leading to eventual failure).
The indirect source of problems from voltage is, as many others pointed out, heat. Heat affects all of the above issues strongly. They would all have an increased effect if somehow the die temperature were a constant and the voltage were increased, but heat is an additional factor. So increased heat leads to these effects occuring more quickly. --pm