My father gave me a family gun when I was 16.
Let's say I was involuntarily hospitalized recently do to an antidepressant overdose. Am I required to turn in my gun?
This is a question with a subjective answer, not a legal one.
"Should I have to turn in my gun"
legally, nobody is forced to turn in their guns because of mental illness. There are fourteen states in the United State that have laws to be able to seize weapons from mentally unstable individuals under what is known as "extreme risk protection orders(ERPOs)", these states are:
California, Connecticut, Delaware, Florida, Illinois, Indiana, Maryland, Massachusetts, New Jersey, New York, Oregon, Rhode Island, Vermont, and Washington.
North Caroline only restricts mentally ill individuals from buying firearms, not from owning them. Please see Possession of firearms by people with mental illness for more information about different states laws on mental illness and firearm use.
I don't know enough to answer this fully, but the federal background check does check for mental health history, though this is not nearly as well documented as criminal background history, so there is that, and involuntary commitment is one of those things that tends to get mentioned. The important issues are the specific nature of your commitment (did a Judge force this on you, or a legal guardian?).
There are several other factors at play, too, as not all mental health issues may be reportable to the FBI (they manage the Databases the Federal Check.). There's also the case that it might be possible to get any flags removed if you are cured. It vary's depending on state to state, but a non-dealer selling or gifting a weapon to someone does not always require a background check, but merely that the person giving the gun does not have any reason to worry about that gun (if your father knows about the commitment, don't ask, it will only get him into legal trouble).
The quickest way to check is to just go down to your local licensed gun shop and ask the guy behind the desk if he can run the background check. It kick back the answer from the FBI immediately, but they can request up to 3 days to make the verification... if they don't see anything in that time, legally, you can purchase a weapon, but as always be careful. If there is an issue with your mental health check, and if you feel comfortable, ask the guy running your check what the rules are about mental health and if there is a way to clear that over time.
The other final issue is that, it sounds like this weapon is a family heirloom of some sorts and depending on the age, it might not qualify as antique weapons have a lot of sales restrictions removed because they're more collectors items than anything else. Again, I don't know specific state laws with regards to this.