Yes, if you keep the WA_iRite folder, the existing iRite instances should still work.
Won't it fail to recognize the iRite instance once the extension is removed. HTML editor did not recognize an iRite instance under my testing user.
The iRite instance should be recognized by HTML Editor. Did you have iRite installed before installing HTML Editor on this other user? Also, what operating system and Dreamweaver version are you running?