You haven't said what kind of computer or monitor you have so I'm going to be asking questions and proceeding from imagination at first.

Do you have the option of moving the CPU and monitor apart to better determine where the smell is coming from?

Are you sure the smell isn't coming from something like a power strip or power plug? Power strips often contain surge protectors which are designed to sacrifice themselves (and trip the house fuses or breakers in the process) if the line voltage gets really excessive. I've had one surge protector fry that way; afterwards it looked like charcoal inside and smelled like an ashtray only worse. When a cord or plug is in the process of failing it'll often feel warm or hot around the trouble spot.

As soon as I got my latest computer (used, 4 years ago) I looked up the user's guide online, printed a copy, and put it in a binder. The paper copy came in handy when my hard drive failed. The user's guide should have information on how to open the case and how to get to the parts that might need dusting.

Most CPUs have forced-air cooling with one or more fans. Fans can fail (and cause overheating) but if you're familiar with the normal sound of your computer you may notice that something is different. I would think that by the time a CPU or one of its "cards" overheated enough to stink you'd notice lots of other symptoms.

I've only ever used CRT monitors. Those have very high voltages inside so they do have the potential to overheat or arc. I know almost nothing about flat-screen monitors.

Good luck!