Plex is fairly open regarding their Internet connectivity requirements. Most apps, particularly smart TV apps, require Internet connectivity in order to load. Here’s some additional information:
The specific setting you mention, “List of IP addresses and networks that are allowed without auth” applies when both the server and client are not logged in to an account.
Having said that, there is a post on these forums which describes how to configure your Plex environment for Internet outages. However, I believe these steps need to be taken before the outage.
