TLDR version: Shield TV problem, not Plex. Nvidia says fix will be in future Shield TV update. Date TBD.
If you have a string of HDMI/HDCP 2.x devices that ends with a HDMI/HDCP 1.x device, then auto frame rate switching does not work.
Example: Shield TV (HDMI 2.0) --> AVR (HDMI 2.0) --> TV (HDMI 1.4)
There's a very long thread on the Shield TV forum.
One workaround is to connect the Shield direct to TV via HDMI. Send audio to AVR via HDMI ARC or optical cable. However, ARC & Toslink do not pass lossless/object audio (TrueHD, Atmos, DTS-HD, DTS:X), so such streams will be transcoded. AC3, PCM, & DTS should pass OK (up to 5.1 channels), depending on capabilities of TV.
There's a way to hack the Shield TV mentioned in the thread, if you don't mind rooting your device.
Thread also mentions using a HDMI splitter. Split output of Shield, sending one HDMI to TV and other to AVR.
Alternate splitter solution is to put splitter between AVR & TV. Have splitter present HDMI 2.0/HDCP 2.2 interface to AVR. Not really splitting the HDMI, just using splitter box to spoof Shield & AVR into thinking TV has HDMI 2.0/HDCP 2.2 interface.
Definitely read the thread if you go down the hack or splitter route. Some posts mention issues with splitters not supporting multi-channel sound or not presenting correct HDMI/HDCP interface to Shield and/or AVR.