An update/correction:
I was wrong: Deleting all cookies for *flyertalk.com did delete persistent login. But it didn't clear the problems getting to the page linked from the big "My Flyertalk" button at the top right: http://www.flyertalk.com/forum/usercp.php
That page and http://www.flyertalk.com/forum/ still do not work for me in this browser. But they work in the incognito window of the browser.
So, I suspect some 3rd-party cookie might be the culprit.