You don't need the TV to be on, or even need a TV at all for that matter! Your Slingbox will stream the video from whatever source you hook it up/configure, and would simply pass that signal through to your TV if you choose to hook it up that way. The remote would be controlling the video source, such as a cable box.
The Slingbox FAQ page may be helpful in giving you an overview of how the Slingbox works: Slingbox.com - Slingbox FAQ
I think this could be a great solution for my need to watch sports at my sister's house -- she has internet but not cable.
The only question is have is whether I need to leave the house with the TV turned on or does sling box somehow let me turn on the tv remotely so I can tune in to sports, etc.