Which TV & STB?
The Plex apps vary widely based on the TV and OS (tizen, webos, android, etc.).
The Plex Smart TV apps (tizen, webos, vidaa, etc.) require an Internet connection to work. The app on the TV is just a small “loader” app. When launched, it checks with plex.tv for updates and downloads the full app if needed. If it cannot check in with plex.tv, then it most likely will not work without an Internet connection.
[HowTo] Use Plex with No Internet , in the section linked by @nibbles, has additional information.
For the most part, you have to have things configured before the Internet is unavailable.
