How do I burn my iTunes purchased TV shows onto a playable DVD?
You probably won't like this answer, but as far as I know, there is no legal way to burn those episodes onto a DVD in a way that is playable in your DVD player. The problem is that the iTunes episodes are encrypted with Apple's FairPlay DRM and DVD players do not know how to decrypt them. In order to make the videos playable in a DVD player, you would need to circumvent the DRM and this is illegal in many countries.

As an alternative, you could buy an AppleTV and hook it up to your television. At $99, it is a relatively inexpensive investment, and it would allow you to stream your iTunes shows to your TV while keeping your laptop and iPad free for your own use.