When the PSU fan dies, what happens next is a matter of luck or quality.
A quality PSU may have either a fan fail shutdown, or overheat protection which will power down the system as the PSU overheats.
A low quality PSU will cook, and burn out, possibly passing overvoltage to the system just before it fails completely - some models are notorious for taking the motherboard with them when they go.
Do not ignore a failing PSU fan - strictly speaking, they are non-servicable units, but so long as you are sure to unplug and be wary of possibly still charged capacitors, opening up and changing the fan is possible (I generally cut and join the new wires to the old, rather than trying to get to the board).
Also, do not confuse normal thermostatic low speed fan operation, with sticking - a simple test (with power off) is to reach through the grille with a thin object and try to flick the fan (this does need care) - if the fan does not spin freely, then it is bad.
Alternatively, take the opportunity to replace the PSU with a better one, ideally an 80-plus efficiency.