OpenHistoricalMap tracks that sort of thing. OpenStreetMap is strictly for what’s currently (and at least semi-permanently) there. For businesses that just closed, it’s a good idea to put a note saying something like “Formerly Dairy Emporium” (along with potentially, say, when it closed if known). This not only potentially helps confused users, but it also helps other contributors who might try to “correct” what seems like a mistake.
Editing and changing the info is ideal, because it preserves the edit history. A deleted object is a lot harder to find the edit history of.