You can close the web browser.
However, the Plex server needs to be kept running. Depending on where you have it installed. e.g. on Windows, the small Plex icon in the task tray is the actual server:

I have the Plex server installed via Synology. Will it run in the background or must I have the app open?
It’ll keep running on the Synology regardless of if you keep the browser/app open on your PC.
