What will happen if my computer power supply's fan stop working?

Most commonly, very little will occur. Many power supplies do not have fans. They are provided so as to assist with the airflow through the machine and that has the added effect of helping to dissipate heat from the power supply, thus leading to longer service life of said power supply but it is not a necessary component of the supply. Having said that, however, if the fan does fail, it could mean that other components of the power supply will soon be marginal and, therefore, you should consider replacement of the power supply in the near future.

The electronics inside of the power supply will overheat and cause the power supply to quit working. There might even be a small fire inside, or the semiconductors may crack or explode. So it is best to repair or replace the power supply when the fan starts to malfunction, since it will be easier to fix. If you must operate it that way, you should take steps to reduce power consumption and make sure the fan is running each time. Some even insert things in there to start the fan if it is stalled, though that has its risks.