You would need to be in a pitch black room with only one pixel at a time lighting the scene. If you want additional light sources in the room they could be photographed and blended in with the same technique.

"You could even set up a small still life around the monitor, and get accurate reflections and ambient light in the room."

I think this would be a final additive pass after the pixel setup. Would also allow you to turn it off selectively...

The more I think about it the more I want to try it out.

I think you would need to get the reflections in the same pass. It’s basically ray tracing in real life: emit light from a bunch of points (screen pixels), “calculate” how they bounce through the room (ie adjust the brightness of the corresponding photo), and combine them all into a single image.