Passing the HDMI signal from cable box through a Slingbox to your TV shouldn't have any effect on the TV's signal, especially if the Slingbox isn't in use (assuming it's powered on).
Give this a shot if you haven't already: Try running the HDMI from the cable box straight to the TV, and see if any similar issue occurs. If you're able, I would also try running HDMI from a DVD player through the Slingbox to the TV, and see if you get the same results, say, watching a movie. That'll help us eliminate the video source as the issue.
Also, try giving the Slingbox a factory reset, and setting it back up if the above does not fix the TV signal issue.
Hope this is helpful!