I had a similar problem with my PS3, the problem turned out to be dirty controllers.
The effect was much more noticeable when I wasn't playing a game as the controllers were not in constant use but symptoms were very similar, random inputs causing the menus to pop up and navigate the menus, usually in one direction constantly.
The simplest solution was to turn the controllers off once you have started watching the show then turn them back on again once its done. I would try this first then you can maybe look into cleaning your controllers with this question or buying a new one if it turns out to be the cause.
Other simple things to do in order to narrow down the problem is check to see if it happens when you are using a specific one of your controllers by recording which of the two is in use when it happens.