A ticket or a chat always has a department associated with it, and neither of them can be created without a department.
In case a ticket department is deleted from help desk, all the tickets linked to that department loose their original department association, and are moved to the 'Trash' folder. Deleted tickets can be checked from :
Staff CP > Tickets > Manage Tickets > Quick Filter > Trash
Similarly, in case of chat logs when a live chat module department is deleted, all the associated chat logs get unlinked from that department, and are no longer visible under Staff CP > Live Chat > History. However, their data remains in the database.