This past week my car has been running hot at idle.

The radiator, fan, and radiator cap are good. There are no water leaks.

The car runs fine on the highway and the temperature is nominal.

I also have trouble starting the car if I shut it off while hot and turn it back on. Starts fine from a cold start.

What could the problem be? Vacuum line? Sensor? CHTS?

The car is a 1985 300ZX Turbo

You may have a bad water pump. My advice is to get it fixed before starting again; otherwise, you could warp the aluminum head and wreck the valves. Hopefully, this has not happened already. Tow the car to the shop, if necessary.

To test for an inoperative water pump, pinch the top radiator hose closed with your hand while the engine is idling, then release it. You should feel a surge of coolant, if the pump is working properly.

If the temp is normal at speed than chances are your water pump is operating properly
If there are no external leaks and coolant level is normal, make sure coolant is cycling from radiator to coolant recovery bottle correctly (it should flow into the bottle when hot and draw back into the radiator as it cools this would indicate that the radiator cap is holding pressure)
Bring the engine up to operating temp with the heater on, shut off engine and feel all radiator surfaces for cold spots. The radiator should be hot all over.if you feel cold spots at the radiator than there is a restriction in the radiator core and it will have to be replaced
Feel all hoses,they should be at operating temp feel all heater hoses too, the hoses should be hot going in and out of heater core. If the hose going into heater core is hot and the one coming out is cool or cold than there is restriction in the heater core
It sounds to me like a circulation problem. As you sit at idle the air surrounding the engine is more static and there is little air movement if coolant circulation was normal this would not be a problem, the fan would pull enough air through the radiator to cool the engine, however with a restriction, not enough coolant would circulate through the radiator to cool the engine, the last clue is the temp is normal at speed, this would indicate the as the car moves, the air circulation around the engine improves enough to compensate for the poor circulation in the radiator