1) When you enable the Debug setting in JFBConnect, there are extra messages that can show up in the system messages output (usually in a blue/yellow/red box above your component area). That may show messages as to why the authentication doesn't work. Those messages are hidden from standard users normally and only show when trying to debug things.
2) Go to the My Account -> My Subscriptions area and double check your Subscriber ID that you entered into the component. It's possible your subscriber ID changed or is incorrect.
3) When you move to a live site, it's still recommended that you re-run Autotune. The recommendations for your Facebook settings are based on the URL of your site, which I'm assuming was 'localhost' on your dev system. Additionally, there are some server checks to ensure there are no configuration issues with Apache or other lower level things that may have changed on your server.
Let me know if you find any error messages when authenticating, and we'll gladly help however we can.
Thanks,
Alex