From a few previous call outs I dealt with for a similar issue of burnt plastic smells, the PSU was the main culprit, usually where its fan has stopped working. The outcome is usually the same if not dealt with: It starts giving off a burning smell for a few days and then the PC will either not turn on or will not power on properly (e.g. LEDs light, but no display, beeps, etc.) In one case where I was present, the PSU made a bang with a puff of smoke. In another case, it shorted out and ended up frying various components on the motherboard.
The only other case I've seen this happen was when the graphics card fan stopped working.
The first thing to check is that the fan is spinning in the PSU. If not, I would strongly recommend replacing the PSU, as usually it is not worth repairing/replacing its internal fan.
If the PSU fan is spinning, remove the side of the PC and make sure all the fans inside are spinning, especially the exhaust fan and the graphics card fan (if it has one) If any of these are not spinning, try dusting them with an air duster, otherwise replace that fan.
If all fans inside are fine and the PC is relatively dust free, then it is very likely your PSU is at the end of its life. In this case, get a new PSU and try using your PC as little as possible, especially since the PSU can potentially cause other damage to the PC.