This is such a massive bug, not having the screen saver kick in at the TV up next menu. I, like you @scottcase have seen so many reports of this problem. I too have had the screen saver start working and then stop again.
The up next menu is buggy as heck and sometimes just freezes at the end of a TV show. The screen saver won’t kick in at that point either.
I pay every month for Plex Pass foolishly thinking I’m supporting the developers for situations just like this! I guess it’s to much to ask for ANYONE from Plex to just drop by and say they acknowledge the seriousness of screen burn-in. Understanding that likely many of their users run $2000+ OLEDs, or heck even a $700 4k TV, and they don’t want to see their software damage a customers equipment, I’m sure. Just an acknowledgement that they will at least prioritize this and need sure to post back before the thread closes.
Plex is specifically marketed and installed on the NVIDIA Shield TV Pro as the best way to organize users media, and even watch TV or DVR shows with a TV tuner. That was the main reason I got the Shield. I’m not a big gamer.
Someone said this issue is happening on the TiVo 4k. Doesn’t that makes this a Plex for Android TV App problem, not a Shield specific problem?
I don’t understand why the Shield doesn’t have it’s own separate Client & Server on Shield, but that’s not the issue, I don’t think.
Side note:
But things like the DVR Subtitles issue. I’ve seen SO MANY people ask about, but never get an answer on this… If I use the Plex DVR (requires Plex Pass) on my Shield with an HDHomeRun Quattro (HDHR is one of the most common OTA & cable capture devices.) After recording a show from from with the Plex DVR OTA, it shows the closed captions as Unknown on play back, and refuses to play them. If I open that same recording into VLC, it plays the Closed Captions without issue?!?
Grrrrr. It’s freaking maddening. Why can the VLC on Shield app play the (unknown) SRT closed captions from OTA Plex DVR recordings, but Plex Client and Server app (all-in-one) for Shield cannot? 
Please, address the up next screen, when playing TV episodes to consistently kick in the screen saver, like my Roku TV does, after a reasonable period of inactivity.
I really don’t want to switch to Emby or Jellyfish. I love most things about this platform. But Plex Pass subscribers who are not asking for fantasy feature requests that will only benifit a niche group need to get a little more attention here.
If this has been addressed and I missed it after searching thread after thread, maybe it’s time to dump the message board format and move to Slack or some other Customer Care platform. Because I have really looked. This had been an issue for a few years.
Thank you in advance.