[adwf]

As a number of people above mentioned, Cookie AutoDelete extension for FF does exactly this.

Sites start out default clearing all cookies the moment you clear the tab. Greylisting clears them when you close the browser. Whitelist retains all cookies.

Multi-account Container and Temporary Container extensions take care of your per-tab container needs.

[qwerty456127]

Although I use Cookie AutoDelete and open GMail (which I didn't whitelist) in a separate container, GMail still remembers me somehow. I have to manually go and clear all the saved data in the Firefox settings to actually reset everything.

[bostonvaulter2]

It sounds like you're hitting a Firefox bug affecting Cookie AutoDelete. Something about the recent changes to cookies has prevented extensions from clearing partitioned cookies.