Overclocking an AVR microcontroller by using an external oscillator with higher frequency than uC supports
The same as overclocking any digital logic device:


its power consumption will increase
this may result in overheating. Lots of overheating will result in physical damage.
above a certain point there will be errors, not necessarily causing resets
peripherals may not work properly (ADC, UART)
small overclocking in favourable conditions will often be completely fine, BUT this has to be experimentally determined for a particular device, speed and temperature


Edit: here's a video  purporting to show an Intel 486 running at 5GHz, ten times its design rating, while being cooled to -196C with liquid nitrogen.