Do the doors on my utility closet need to be vented steel?
We have the same arrangement in our 1970 house with slab on grade foundation--both water heater and furnace natural gas fired in a utility closet centrally located on the living floor. All of the houses in our Fox and Jacobs 200 house tract neighborhood are built that way.

These gas fired heaters get their combustion air from inside the closet which is open to the the attic via wire mesh in the ceiling. The double doors to the living space are standard hollow core doors, no vents, and seal fairly well.

The conditioned air is pulled into the furnace air handler through a plenum below.

In your case is there a wire mesh over an opening into the attic? Is the attic vented (soffitt and ridge)? Or do your water heater and furnace get their combustion air from outside via roof or soffitt vents?

It would be highly unusual for the WH and furnace to be getting their combustion air from the living space, but if they did then you would need a grill in the door, otherwise no.

But find out what is code in your jurisdiction.

EDIT Another possibility: Where does the air flow into the air handler enter the closet? My parents had a small, nicely built retirement home in which the air entered the air handler through a vent in the door. The door when shut sealed against the inlet to the air handler. If you have this arrangement, the door must be vented. In my house the air enters under the door through vents into a plenum and then goes up through the filter, the fan, the furnace and the a/c evaporator