Can I use the amazon fire tv stick as a way to record live tv with plex dvr
Record, no. View recorded shows, yes. View live TV, yes. In the past the Fire TV stick was not very powerful. A better choice is the Fire TV box. Much more grunt. You either need to transcode the shows using an HDHomeRun Extend, use the experimental transcoding on HDHomeRun Connect, and/or have a powerful enough server to do the transcoding on-the-fly.
@davidm30 As I believe the Fire TV devices now use the standard Android TV interface, I think you should now be able to schedule recordings on your Media Server using that device. 
