Running an engine without coolant forces the oil to act as a coolant as well as lube. It can not transfer the heat as well as water. Areas that will take the brunt would be the cylinder bores if the pistons thermally expand and score the cylinder walls and the excessive heat can take the temper out of the piston rings. Also in the head the excessive heat can cause the valves to expand in the guides and score. Valve stem seals also can be pre-maturely aged and may start to leak. A sever over temp will age the oil and give it a funky smell.
Damage would be noticed by hard starting, low compression or by white exhaust indicating burning oil. ( oil level change also)
Extreme over temps will also affect gaskets and bearings and warp heads. We had a small block chevy on an engine dyno running a thermal shock test run at WOT without coolant due to a dyno controller dying. The engine was stopped when this was detected. The oil stunk but the engine still started and ran when the oil was changed out after it was allowed to cool.
Cal 9.2 #19 SilverSwan