This is not a duplicate question about websites blocking TOR by IP. I'm hitting some issue where clicking on the login button or dropdown with Disqus and other commenting sites opens a new browser tab instead of opening a popup window on the same page. Then, though I'm able to login successfully in that new tab, it simply disappears after I login, and back on the original page it still thinks I'm not logged in - reloading the page doesn't fix it.
I first thought it was a cookie problem, but even enabling 3rd party cookies by setting network.cookie.cookieBehavior to 0 doesn't fix it. It seems like the issue is whatever's forcing a new tab to open instead of allowing the inline login popup - though that's not the whole picture, because with normal Firefox, if I login via one page, I'm considered logged in on all other pages as well, even in private browsing mode.
And to answer the expected question about logging in "defeating the purpose" of using TOR, no not if everything - an anonymous email, the messaging account, etc. was creating in the same TOR session with the intent of being thrown away after that use.