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Firefox becomes slow after a couple of weeks open
(bugzilla.mozilla.org)
A place to discuss the news and latest developments on the open-source browser Firefox
What you're describing is called a resource leak. Something, an extension, a background process, etc., is holding onto resources for too long without cleaning itself up automatically.
This is pretty common in writing code, and extremely difficult to prevent except in closed and well understood systems. A browser is anything but that, due to the nature of needing to work on any website doing whatever they want.