Is it possible to run a small circut off the power that is output from a unpowered headphone/speaker?
In principle it could be feasible if you are crazy micropower everything and using techniques like energy harvesting. However there is one tiny little issue:
You will inject noise in the audio path
Every time you suck a little power, that would create a voltage across the wire that will be read as audio signal by the receiver. So the result would be probably some interference like a cell phone in the tv audio.