Known Issues with Lightroom and/or my plugins...
Spring 2013 — Facebook
Facebook's servers for third-party apps like the plugin are remarkably flaky, and many users get upload errors at random times... sometimes with every upload. There's nothing we can do about it but hope they get their act together, or move your photos to a better photo-hosting site.
However, if you're having trouble authenticating, make sure you're using version 20130330.225 (or later), as that has an update to track a change in Facebook's authentication procedure.