I saw it in "Jurassic park" movie. Character of Sam Jackson says "Vehicle headlights are on and not responding. Those shouldn't be running off of car batteries".
I found this version:
Vehicle headlights are on and they’re not responding. Those shouldn’t be running off car batteries. Item one fifty-one on today’s glitch list. We have all the problems of a major theme park and a major zoo, and the computers aren’t even on their feet yet.
The person in the movie says that the headlights should not rely on car batteries. They should use the centralized source of power, but for some reason they do use car batteries as a source of power.
I found this definition in the American Heritage Dictionary of Phrasal Verbs:
- To operate using something as a source of power: This CD player is portable and runs off batteries.
There's a whole page dedicated to the explanation of this "glitch" in the movie.
As to why it could be "off of", I'm not sure, but I do sometimes come across such combinations. Here's a question over on ELU SE about "off of". It might be an issue of style.
In your example, the overall meaning is the same for "off" and "off of", IMHO.