What can happen if the computer fan do not function?

Depends on which fans you're talking about. The mobo and CPU will have the safety thermal measures startup and shutdown the rig when it over heats or detects faulty fans. If that does not happen then you can end up facing a component burnout.

And if it's a PSU fan not working, then you had better watch out and get a new one immediately, since overheating in the PSU could cause current to leak into the mobo and other peripherals attached to the PSU, permanently damaging them or burning them off.

Sorry, I just realized you've posted it under the GPU section, which most probably implies your GPU fan's not working. In that case, it could overheat the card causing, component burnout, card bending and basically everything that could go wrong when a card catches fire. And of course the system rebooting under heavy graphics load or games crashing, desktop getting corrupted etc etc.

generally overheating issues and shutdowns or not booting, sometimes you get something burning.

also depends on the fan, if its a case fan then it isn't too big a deal but for the cpu fan, it won't work if you don't have a fan.

You can minimize the possibility of damage to the CPU at least by enabling temperature warnings in the BIOS. Go to the PC Health tab and set the CPU warning for 60 C (AMD) or 70 C (Intel).