What happens to an LED if you exceed the maximum current?

Exceeding the maximum current of an LED can cause it to fail.

Increasing heat from power dissipation causes a failure of the LED die.

The change in color, e.g. red and green LEDs going yellow at high currents, is probably because the die is actually glowing hot, i.e. near failure.

Note that the red LED has a fall in wavelength, but the green one has an increase in wavelength. White LEDs going blue could be explained by the yellow-emitting phosphor in the LED being less effective at high currents.