After a long time, I can finally confirm it is working on macOS and Firefox with a cookie whitelist (cookies disabled by default, only user-specified pages are allowed to set cookies). Thanks to the developer to create an actual support document for this edge case.

If you are a Firefox user who likes to whitelist cookies instead of accepting everything by default, here is your guide:
http://rescuetime.helpscoutdocs.com/article/90-the-rescuetime-firefox-add-on-does-not-work-with-my-custom-privacy-cookie-settings-in-firefox