Maxvorstadt: Maybe CD-/DVD-Drive?
mechmouse: That would be impressive. Its in the attic, waiting for the day I need to use physical media again (ripped all our discs long time back).
Its madness, I'm struggling to think of anytime in 20+ years of IT I've come across something like this. I might buy a high gain mic just so I can pin point what's making the noise.
This is starting to pique my interest now that you've ruled out most of the moving parts in the box. You've only ever encountered this when running Project Hospital? Never with anything else? That being the case, maybe run perfmon counters in a comparison to see what kind of resources the app is using as compared to another something close in resource utilization. That's a long shot, but more data is better than less.
You don't have anything custom in the case that would mean additional mechanical parts like a pump for water cooling?
Just one other thought. If you are using windows 10 there are sound settings for individual apps and audio output devices depending on the release you're on. Could be that a sound setting for the app is overriding the system sound setting and you aren't actually muted. Another long shot, I know, but I'm just spit balling here.