Most modern diesels do not, since they can run the prime mover at very low speed to just power the HEP. The HEP is fed from an additional inverter on the main DC bus, the same one that has the traction alternators hanging off of it. The power demand placed on the bus determines how fast the prime mover is run to get the alternator to produce enough power to satisfy the demand. When there is low power demand the prime mover runs only as hard as it needs to to satisfy that demand. Wonders of solid state power electronics. An added advantage is regenerated power feeds the HEP first before feeding anything to the dynamic grid or battery or whatever one does with surplus power in the locomotive. Of course in elecrics the surplus is fed back to the catenary if it is receptive.
And of course they are also Tier IV prime movers.
In the P42 generation, there is a special HEP only non traction mode, which essentially does the same thing. That is why when at a stop it is determined that the stop will be long, you momentarily lose HEP power as they transition the engine from traction mode to HEP mode. And then just before starting to move again they change the mode to traction momentarily switching off HEP for the transition.
In the F40 generation unless HEP was provided by a separate pony motor, it was the prime mover that was screaming away at a stop providing HEP and waking up even the dead within miles of the stop.