As I understand, zero fuel weight refers to certain aircraft (think larger wing spans) that must have a significant portion of the useful load as fuel.
Weight in the form of passengers & cargo, even if they are well below MGTW, place excessive stress on the wing in turbulent conditions.
Said another way, an example aircraft may have a useful load of 1000 pounds, but 400 pounds MUST be as fuel in order for the aircraft to be safely loaded. Meaning only 600 pounds may be people & stuff. Go over that number and you exceed the zero fuel weight.
IF an aircraft has a zero fuel weight requirement, it will be listed in the POH for that aircraft.
PP studying for CFI