As far as I can tell, this issue started after I upgraded to Sierra last week.
My Plex server is more often than not inaccessible on my iPhone and Roku TV, whether accessing it remotely or locally. My current theory is it becomes unreachable after the server machine goes to sleep. When I wake the machine from sleep, my Plex devices can see and connect to my server again, but even then, sometimes it can't actually play media. If I'm sitting in the same room I can hear my external drive spin up, as it's attempting to access the media, but the app never manages to play it. Incredibly frustrating.
Sometimes it work, but it's few and far between right now.
Running the latest PlexPass version 1.2.1 (was having the issue on 1.14 as well) on 10.12 Sierra on a Mac Mini (2012).
EDIT: Just got home, and saw an update to 1.2.2 was available. Applied the update, and the issue still wasn't fixed. Even with my computer awake and my iPhone on the local network, I couldn't load media.
Then it dawned on my I hadn't tried rebooting the Mac Mini. After doing that, it seems to work now - even if I manually put the Mac to sleep and access media from my phone.
Will keep an eye out to see if it's fine now, but so far the reboot seems to have done the trick.