With the help of Google, I was able to piece together a solution for preventing my Linux computer from sleeping while streaming something via Plex.
My workflow right now is to use an app called Wake Me Up to send a wake-on-lan signal to my Linux computer running Plex Media Server, which allows my Plex client apps to discover the server. The server won't go to sleep while there are any streams running from the server as long as the service defined below is running.