Eh, I doubt they would worry about it. Astronauts can hardly take a piss without mission control having planned it out beforehand.

Streaming video of a preprogrammed autonomous vehicle is about as close to "Reality TV" (read: scripted and/or heavily edited pseudo-organic drama) as it can get, eh? Honestly, if something unexpected "slips" out in such a case, I'd think they'd be thrilled; it's an opportunity to spin it into real drama (a problem in need of a solution), or it's an opportunity to show the scientific process, a teachable moment (sometimes unexpected things happen; sometimes tools break, sometimes you discover aliens, you know, whatever). Any producer would be thrilled to have their own microcosm of Apollo 13 (the movie), right?
Mostly though, it'd just be supremely boring... these are not powerful vehicles, travel is almost as fast as paint drying. I mean, they can still give a schedule for expected highlights ("drill arm will be active tonight at X, check it out"..), and there'll always be a few people watching any time day or night, and sometimes they'll spot something odd and get some Internet Points from it.
Heck, maybe in say the next decade, we'll be far enough along that this sort of thing is actually feasible, who knows. Or if not on Mars, then on the Moon surely, or when they get down to mining asteroids perhaps...

Tim