Plex Media Server as a whole will only update when you install a new version of the application (whether manually or initiated from the update check in the Plex Web App). However, channels, plugins, and some other server components will update automatically on their own.
What and When
Items within your Server's "Plug-Ins" directory are able to be dynamically updated. Since Plex Media Server version 0.9.12.5 and later, this basically means channels and other plugins.
Your Server will check for updates regularly. When new versions of a component are available, your Server should find the update within a few hours, then automatically download and apply the update.
Related Page: How do I find the Plug-Ins folder?
Other components can be downloaded when first required in some cases, such as when playing new types of content for the first time. After the initial time, subsequent playbacks do not require an internet connection.