I am pleased to report that the Plex Media Server (PMS) addon that hugepants has been maintaining over on the OE forums works perfectly with LibreELEC.
As of August 30 2016, version 1.1.2 of the addon (PMS 1.0.3) works perfectly on LibreELEC 7.0.2.
To review, Plex Media Server is a UPNP/DLNA server with transcoding support and iOS/Android/PC/Mac/browser clients.
To administer the server after the addon is installed, simply point your web browser to the the following URL:
Users should replace 192.168.1.100 with the IP address of their LibreELEC machine. The IP address of your LibreELEC machine can be found under the System -> System Info menu in Kodi.
More detailed instructions on how to install and configure the Plex Media Server can be found over on the Plex Support Site.