It is a problem cause by a change in Facebook. Facebook deprecated their Graph API version 2.2 which is the version supported by Moodle's leading Oauth Plugin (https://moodle.org/plugins/auth_googleoauth2).
The Plugin description clearly says the following
Warning for Facebook users: I have been alerted that the current install of this plugin will stop working with Facebook API 2.2 the 27 March (it was already not working for the facebook API 2.8 and later). Unhappily I do not foresee any fix to this issue. However I will be happy to merge a pull request.
Facebook not working anymore
Facebook changed the API in its 2.8 version and it is not compatible with the plugin. Moreover the plugin is compatible with the old Facebook API 2.2 which is ending the 27th of March 2017. So it is likely that the Facebook login will stop to work at this date. In the comments you will find a script to put in your root folder to detect who is using facebook and so you can alter your users.
Although certain other alternatives are given for this plugin in the description page itself, none of them seem to be working with the current facebook Graph API version 2.8.
We have a login system that allows login only through Facebook and hence all our users have been logged out for more than a day now. This is a serious issue for us, sadly no one is able to help out.