I'm kind of new to org-mode in Emacs, and I'm investigating if it is suitable to replace Google Tasks for me. Google Tasks offers a "clear completed tasks" action, that removes all tasks that are marked done. Can I do something similar in org-mode with DONE items?
Yes in org-mode you do archive DONE tasks (with in the same file or to a subling file) . you can use C-c C-x C-a for that.
please refer the manual http://orgmode.org/manual/Capture-_002d-Refile-_002d-Archive.html