If your default browser is already open before you ask Lightroom to authorize, does it work then?
I've tried it with the browser closed first, then with the browser open, and then with the browser open and logged into Facebook. Each time there seemed to be no connection between Lightroom and Firefox.
Did you get it sorted yet?
Was this every fixed? I am having exactly the same issue. (From "Facebook Set-Up", I click "Authorize on Facebook", then click "OK", and nothing happens.)
I made some progress with Jeffrey Friedl's third-party app, but am getting 'System returned “securityError” when Lightroom attempted to use the Internet: Are you sure that your system date and time are correct?'. Have read Jeffrey's FAQ (Jeffrey Friedl's Blog » FAQ for Jeffrey’s Plugins for Adobe Lightroom), but system date/time are set correctly, and if there is something blocking Lightroom's access to the Internet, I have no idea what it might be (have checked System settings and Chrome settings).
FYI--Working with Lightroom 5.7 / OS X 10.8.3 / Chrome 42.0.2311.90 (64-bit).
Any help would be greatly appreciated. This feature is so key to me that I may stop using Lightroom entirely if I'm not able to solve it. Thanks in advance.