Plex Media Server as a whole will only update when you install a new version of the application (a manual installation, one initiated from the update check in the Plex Web App, updating an NVIDIA SHIELD install on Google Play, etc.). However, there are some other server components and information that will update automatically on their own.
When you launch Plex Media Server (and then occasionally while it’s running), the server will contact some cloud services on plex.tv to update information. This can include account information such as subscription status or connection information. Information is also gathered related to access to certain features or functionality. For instance, this allows us to release server updates (and updates to various player apps) that might offer support for an upcoming feature and then turn on that feature at a later time, when it’s ready to be launched for users.
In other cases, components can be downloaded when required, such as when playing new types of content for the first time after updating Plex Media Server. After the initial time, such subsequent playbacks do not require an internet connection.