Justin,
I'm not sure either, actually. I re-read your description of the problem, and agree your issue isn't want Q2/Q3 of Common Support was meant to address.
Usually, when the login box doesn't go away, it's caused by:
1) Incorrect communication between your user and Facebook. We try to address this by creating a 'channel', recommended by Facebook, so that everything stays in sync
2) Other Facebook extensions are interfering
3) Browser specific - This almost always occurs in IE more than any other browser, and many times, won't occur in the others.
4) A general Javascript error on your page, possibly, but not necessarily related to Facebook.
Other things to test to see if you can narrow down the issue:
1) Disable any other Facebook integration extensions you have enabled. Mainly, anything that may be trying to do registration, but even other Like buttons, etc can cause issues depending on how they're implemented.
2) Try a different browser or even computer
3) Try a different template on the site, preferably one that ships with Joomla (beez5, atomic, etc)
4) Try placing the JFBCLogin module on the page, not in a Javascript pop-up
If any of those fix the issue, let us know, as that will give good indicators as to where the underlying problem is and what to do next. Additionally, feel free to post (or Private message) your site's address, and we'll take a look at the page to see if there's anything else obvious going on.
Thanks,
Alex