Thanks for the logs. They do show about 200 requests timing out - and during the same period about 1600 requests were successful. It goes over a number of hours - so it is not failure due to short outage. It could be some network issue between your server and plex.tv systems or problem with your router.
Wireshark capture would help here and I can look at it if you zip and send me privately and let me know the time period covered by the capture.
I suspect you are running Tautulli and it is making requests to plex.tv every minute to check remote access. See this forum topic for discussion on that Random loss of remote - hundreds of pubsub async errors
If there was any rate limiting being applied (due to the volume of requests), I would expect a specific response code rather than a timeout
