If you cosigned on a year lease on an apartment for your son can the apartment hold you responsible when the lease is up if he doesn't pay his rent?

Yes. anytime you co-sign a lease you are 100% responsible not only for the money of rent but 100% of all other terms. Such as damage, late fees, early termination fees, court costs, etc. for the full duration of the family members tenant status. In a lease or month-to-month the co-signer remains responsible until the tenant is no longer on property. If the co-signer wants to be relieved of future liability, then the co-signer should send a written notice to the landlord, before the lease renews or goes month-to-month, requesting he/she to be removed from the agreement as a co-signer. I would imagine the terms and rules would still apply, even though the lease expired and is now month-to-month. So write a letter and give notice that you are ending your arrangement.