It depends, if it overheats, some power supplies have OTP (overtemp protection). I guarantee you that your AX1200 has this protection. On the other hand, most cheaper power supplies don't.
As an example, where I work we are in the midst of replacing our old Windows XP computers. I recently came across one of our old Optiplex 320 SFF PCs, which has a Pentium 4 Prescott in it. In addition to the Nippon Chemi-con KZJ capacitors on the motherboard puking electrolyte all over themselves, the hipro built power supply actually had a completely seized fan, and it had probably been seized for a while. This computer had been run quite a while this way it seems. The heat of the Pentium 4, plus the lack of fan cooling the PSU, meant that when I opened it up, every single capacitor in the power supply was bulging or leaking. Granted, this was an old computer, so even if the fan hadn't failed, it could still have had a few bad caps regardless. But the point is, despite the fan failing, it continued to run until we replaced it.
This won't always happen though. In a lot of super cheap power supplies, if they overheat, you will see fireworks under load, when the primary switchers explode. Hardware insights has a few good video examples of this happening under their el-cheapo psu roundups.