I don't think so. The way the system is made the background app is completely paused when you press the home button. If the app is doing anything that can't be paused it won't let you go to the home screen. Unlike, for example, the PS3 where if you press the home button it places an interface over the the app (even in the same resolution as the game you are playing, which is strange).
To be honest, what device lets you to do that, PC notwithstanding? Even with smartphones you have to be on the app to keep doing ****. The closest thing that comes to what you are talking about is being able to watch netflix/youtube with a chromecast and continuing to use your laptop/phone after it was pushed to your television.