I never really understood Boeings "Engine Overheat". I have idled or shut down RR engines on 757 4 times due to Engine Overheat. But it really has little to do with the engine itself, but is a bleed leak in the Strut (pylon if you will). As it is upstream of the Bleed Valve, switching the bleeds OFF does nothing. Having really hot air for long around fuel and hydraulic lines is not a good idea so the Boeing procedure is to idle the engine, and if the warning goes away after some cooling time (Boeing does not specifiy, used the 2 min they suggest for engine shut-down ) keep the engine running at idle, or if warning still on, shut it down.
It has been a long time but on the 737 Classics IIRC it was the same thing.
It will not show up on your engine instruments in any way.
Spurious warnings are a different thing though, probably a wiring or a plug issue. Turned out, my events were all actual leak.