My iMac M3 is overheating and the fan stays silent

I would, yes. Disconnecting a power cord is in the "abundance of caution" category of advice but it's not a bad idea.

Before you make an appointment with Apple run Apple Diagnostics: Use Apple Diagnostics to test your Mac - Apple Support. Among other tests it will exercise the exhaust fan and confirm proper operation. Most likely, it will return "no trouble found" but any other outcome should be brought to Apple's attention.

Regardless of its outcome a service appointment is justified: Official Apple Support

I'm aware of the heating issue, with iMacs and I only wanted to put the graphics down to suit my iMac

I don't quite understand those references so if it is relevant please explain.

I understand your concern. "Hot" is of course somewhat subjective, and the iMac incorporates many features in both hardware and software to prevent a real i.e. potentially hazardous overheat condition — it should spontaneously shut down on its own before any damage could occur — but personally I would be uncomfortable offhandedly dismissing symptoms described as "really hot with the burning smell." What's next, smoke? That just isn't right.

Maybe it will never happen again...

Apple Diagnostics will exercise the fan, and will report a failure to meet its expected speed. However, the nature of your concern suggests the problem was either a singular failure event that may never happen again, or one that may be intermittent and could return. Problems like that can be frustratingly difficult or even impossible to conclusively determine.

If you were to have Apple examine it, in all likelihood they'll reach the same conclusion as Apple Diagnostics. Don't let that dissuade you from taking it to them though. I think it's justified, and it's possible they might find something wrong. Better to know than to remain concerned.

niclas6573 wrote:

Still, I probably can't (don't dare) to play this full version of the game anymore with this Mac? When I bought this Mac, I was really confident without checking properly, that "Sure, of course this Mac will handle the game perfectly, without any kind of issues".

I would be just as confident. No doubt gaming is a resource-intensive task but you can't hurt a Mac just by using it.

I'd continue to play that game, full version, just be mindful of it happening again.

I wonder, when they have this "lighter iPad - version" of Football Manager on Apple Arcade, that if that version is compatible with my iMac? After all, it's the "Apple Arcade".

That I don't know.

niclas6573 wrote:

During this short session the iMac started to overheat that bad that it even was starting to smell burned!

This concerns me, because a hot running computer rarely has any smell unless the system is full of dust or some foreign materials (including remnants of a liquid spill such as coffee or milk....which will smell when heated).

The only other time you will have a burning smell is if a component on the Logic Board has shorted out causing that component to burn out. Usually a burnt out component is fatal & the device won't function properly, but some times it may not show any noticeable issues. Usually these components will "burn" for a few seconds then burn themselves out as the shorted line becomes disconnected from the circuit.

The other remarkable thing was of course that all this time the fan was staying silent. Normally it would have turned on and the iMac wouldn't have turned that hot like it did.

That is the new normal with the M-series Macs. It takes a lot to have the fans run at high speed where you can actually hear them. If the Apple Diagnostics are not reporting any fan or temperature sensor issues, then the cooling system is functioning.

Should I bring it to the Apple service?

A burned out component will leave a faint odor for a while. If there is a burnt component, then having Apple examine the laptop now may allow them to actually smell the burnt component to confirm. They can also tell if any accidental liquid damage is involved.

Now I will say that if a forum moderator sees your post, they will highly recommend you have your laptop checked out by an Apple Store or an Apple Authorized Service Provider because of the "burning smell" you noticed since Apple wants to make sure the laptop is not a hazard to your or your home.