The hot water baseboard in my apartment stays warm when the heat is off. It is working but does not completely turn off. The head on the zone valve has been replaced. Is this normal or should the base board shut off completely.
Depending on the pipe layout between the baseboard and the boiler, it's not too uncommon for the baseboard closest to the boiler (typically) to have an inadvertent thermosiphon when off, which can cause the temperature to rise above the setpoint of the thermostat.
If the temperature is not rising above the setpoint, see @ratchet freak's answer and @JACK's comment
Water has a very high heat capacity. This means that, after the hot water stops flowing when the heat shuts off, it will remain sitting in the radiator radiating heat into the room.
This is by design and why a thermostat tends to shut off the heat a few degrees early because it expects an overshoot.
hot water baseboard in my apartment stays warm when the heat is off...