I'm usually too lazy and I just leave it where it is and don't do anything about it, but when I'm not lazy I usually just throw them away. Sometimes I'll sell them/give them away here if I think the items would be useful to other people.
Hoard them until there's so much that I just can't keep it anymore, then sell it all to Reese and end up making like 200,000 bells. Is being a hoarder worth 200,000 bells? Idk