I'd start by taking a look at all the fans in your system, including the GPU and PSU fans. A dying fan might cause a buzzing sound at certain RPMs. Some games might cause the fan to spin at that RPM more than others. You should be able to identify if this is the case by playing with fan curves.

Your GPU might also have coil whine. Coil whine only shows up when the GPU is doing something and it depends on what the GPU is doing too. Some games may cause a lot and others will have none. It will also come and go depending on what's going on onscreen.

There's not a lot you can do about that but coil whine in the GPU can be effected by the PSU. If you have two power connectors on that GPU make sure you have two separate cables running to it. If you have a modular PSU you can also try switching the GPU power cable to another connector on the PSU or try different cables. Swapping the PSU entirely may also help. Worth trying if you have a spare kicking around.

Coil whine can also be effected by the power quality from the wall. You might want to try plugging in the computer directly into the wall if its on a power bar or trying it somewhere else in your house. A good quality uninterruptable power supply (UPS) might also help if you have one to try.