You should check what is happening on this server on Sundays. Maybe some scripts, group policy, backups, etc. Openfire shouldn't just turn off on Sundays on itself.
So you are logging into a server and run Openfire with the Launcher? Then you leave it this way running without loggin off? What operating system? Are you logging via Remote Desktop. I haen't heard about such thing, but probably there is some timeout for idle login sessions (maybe even configurable). Maybe it just terminates idle sessions after 7 days. Then it should be enough to login after a ~5 days into the same session and check if it will live through the next Sunday. Just to test this theory.