My bathroom door isn't latching properly and it seems that the strike plate is not seated properly in the jamb. It's a 1950s apartment and all the door frames are metal, so I can't just chip away some wood. What else can I try?
The strike plate sticks out from the jamb? Remove the strike plate and file/grind some material off where it is binding.
Sometimes the hinges might not be where they were when it worked correctly. Make sure all your hinge screws are tight and not stripped out. this can make the door sit off kilter. Check your gaps around the door between the door and the jam. this means compare the top to the bottom and left and right. They all should be about the same distance. The first answer is correct also, but check your hinge screws. It could also be your problem.
The strike might have been replaced at some point. If you have other doors in the apartment that work properly, take that strike out and compare it to the one in your bathroom. That will tell you what modifications need to be made to get the door to close. If you have a closet or other door you don't worry about latching, just swap them out.
If you rent the apartment, then the landlord is responsible for maintaining and repairing the doors and locks. If you own the apartment, then you're responsible. Since the door jamb is steel, filing the striker plate to a larger size is your only option as long as the hinges are not causing the door to sag.