A while ago Plex added live TV and DVR features to Plex on Android and iOS devices and now they have added it to Amazon Fire TV and the web. You will be able to use an Amazon Fire TV stick to watch live TV or recorded TV from your Plex server, you will also be able to use the DVR and live TV features via a browser.
For the server you need a Plex Pass and a support TV tuner. More info on live TV at Plex.tv