Facebook - AutoTune issue

I'm unsure what's happening on your site. I've never seen that behavior before. When I try to login in the popup, the popup never shows the authentication form like it should. Instead, the popup is empty/white. Is that what you're seeing?

I'd recommend trying the following:
1) Create a new Facebook Application in the FB Developer's area and use that API/Secret Key. That shouldn't be necessary, but sometimes the apps can be configured wrong and cause issues.
2) Change the "Show 'login credentials' in popup" setting to "No" in the JFBConnect -> Configuration -> Facebook area. That will not use a popup for credentials and instead redirect to Facebook.com.. it may help provide more information about what's going wrong by logging in that way.

I hope that helps,
Alex