I haven't researched this, but I believe the data rate available from most space probes is insufficient for a live picture. Certainly this was true for older probes. There may be methods around this eventually. For example, there have been proposals to place large communications satellites in Mars orbit. Such systems could raise the bandwidth to the point where a webcam-type data rate could work.

That said, I think live cameras on probes and other space vehicles are unbelievably cool. The idea of watching the camera on a rover and knowing that you (okay, and a few others) might be the first to see something amazing. I would pay good money for such a hookup.