My mother lives in NYC and has a new hospital bed. How or where can I donate the older, serviceable bed?

I would call Housing Works and see if they will pick up.

Call a local Nurses Lending closet. Senior Center or if you have an Adult Day Care in the area they may know of someone that needs a bed.
Placing an ad on your FB Marketplace or FB Garage Sale might be an option.
Contacting Salvation Army or other Resale shop. If they can pick up.
Last resort would be to place the metal portions out with the trash and scrappers will pick it up and it will be recycled.
NO ONE will take the used mattress so that can get tossed out