I have an older shotgun designed for lead shot, by law I can only fire steel shot.
Can I fire steel shot, or do I need get a new shotgun?
If I can use steel shot, what additional maintenance is required?
You shouldn't shoot steel shot through a shotgun barrel that isn't designed for it. Steel shot builds higher pressure than lead and could seriously damage both your gun and your face if the manufacturer didn't design for it. This kind of warning will be written on any box of steel shot ammunition you buy.
You say "by law I can only fire steel shot", but is steel specifically mentioned or does the law require "non-toxic"? If non-toxic you have other non-steel options but I don't know if other shot types carry the same risk as steel.
Your best option by far is to ask in a gun store, preferably one with an in-house gunsmith. They'll likely want to sell you a new shotgun regardless but should still give solid and safe advice.
In addition, it isn't only the gun itself you should worry about with steel shot. For instance, my turkey plug is only for lead shot. I believe my shotgun can take either, but the specific plug I am using for turkey specifically said no steel.
For additional maintenance, I don't think there would be as much extra lead residue in your barrel (obviously), but you will probably find more residue from the powder to get the higher pressured shot out.
Check with the manufacturer of the gun to see what they recommend for that particular model. If is it really old, then definitely consult a gunsmith or someone whose job it is to know more about guns than us common folk.