Do I need hook_update_N if there are no database changes? I wrote a little module that provides a form and does some special processing of the submission.  There are no database tables involved. If I change the form or processing, how do I code the update? Do I still use hook_update_N if the module update doesn't affect the database?

Thanks!
If the changes are just code, (not db schema changes, content changes, variable changes, data changes in general), then you don't need an update hook, no.

Basic rule of thumb: will your code updates break anything for existing or new users? If not, you don't need an update hook.