After recently updating to LR 6.4 In publishing services LR will not authorise Facebook. Can't believe there is no solution available !!
Could you please let me know if you were able to publish to Facebook prior to the update ?
Also which computer are you working on , Windows or MAC ?
Do you get any error when trying to Authorize Facebook ?
I am having the same problem. Mac El Capitan.
Are you also using the most recent version of Lightroom, 6.4 ?
Also did you try to remove and then reauthorize Facebook ?