What will happen when the current is too high for a switch?

Switch damage due to excess current goes one of 3 ways:

1) If the damage occurs as the switch is opening (which is the common failure condition when dealing with motors and solenoids) the switch contacts burn out, and the switch will no longer close the circuit.

2) If the damage occurs as the switch is closing (which is likely when switching resistive or capacitive loads) the contacts may weld together, and opening the switch to cut off power becomes impossible.

3) In either case, a middle ground sometimes occurs, and the switch behaves as a (relatively) high resistance - that is, high enough to dissipate considerable power but low enough not to limit the current appreciably. In this case the switch may catch fire or blow up.

Switches that specify their lifetime (ie the better quality ones) often say something in the data sheet like ...

mechanical lifetime (no load) 10,000,000 operations
electrical lifetime (rated load) 100,000 operations

unfortunately the lifetime does not change linearly as the load, but usually exponentially

a fictional data sheet might continue ...

overload lifetime (120% load) 1000 operations
overload lifetime (140% load) 10 operations

but of course no switch manufacturers warrant their switches above their rating, with good reason.

You might also bear in mind the effect of the load. Resistive loads (like lamps) are much tamer than inductive loads (like motors). When switched off, an inductive load will try to keep drawing current, raising the voltage across the opening contacts to maintain the arc. Switches will often specify a difference in lifetime of a hundred times between resistive and inductive loads of the same current.