Dropping a ring of hunger into a sink produces the message, "Suddenly, <item> vanishes from the sink!" but only if there's another item on the sink. Otherwise, "You hear the ring bouncing down the drain pipe."
The ring of hunger is not an entirely useless ring, but would only be worn in rare circumstances. If, for example, one wanted to eat multiple giant corpses without an Amulet of magical breathing, you might wear this ring to make more room for the meal. However, the ring of regeneration and the ring of conflict also convey hunger while actually doing something useful, making this ring primarily used as polyfodder.