I saw this video some time ago:
How may I know if a chip that needs cooling will reduce speed if temperature increases (maybe because the fan breaks)?
What about current CPUs and GPUs?
GPUs seem to do the same thing: see here http://www.pcstats.com/articleview.cfm?articleid=1803&page=7
You will definitely notice when it is running to hot. As a consequence of being to hot the electricity inside the chip will start to jump around randomly, and thus causing data to switch positions etc. This will cause all kinds of weird errors and bluescreens.
Fortunately all current generation chips will automatically run at a lower clock speed to prevent it from running to hot. They will even shut themselves down when necessary.
On top of that, if your CPU fan fails you will hear beeps coming from the motherboard.