Hello and welcome to Apple Support Communities justinfromnorthglenn,
Thank you for contacting us regarding watching TV shows on your Apple TV . You have come to the right place and we are happy to assist you. If we understand correctly, when you watch one of your TV shows from "Up Next", it doesn't continue playing when one episode ends. We can understand why you’d reach out for a solution. To start, does the TV have the most current operating system? If so, then if you can give us a bit more detail on what your experiencing, we could assist better. What App are you using to watch? Is it downloaded to your library or streaming?
Thank you for contacting Apple Support Communities.
Best regards.