If you change username and password from the original session to one which will not authenticate, Firefox will destroy all reference to the old authentication and attempt the new one.
You can change/set authentication by adding user:pass@ to the beginning of the URL, example:
If you are logged into www.example.com with user:pass then typing (adding to the beginning of the address) anything@ will cause this to happen, example:
A fast fix which doesn't require any plugins.
Please note this only works for HTTP Basic Auth.
An additional side benefit is this will affect only the website you are interacting with. All other logins to any other website will remain active.