As a physicist I would very much doubt that the brain will burn significantly more calories when listening to music. I'm not saying that you wouldn't be able to measure something with very sensitive equipment - but it won't be enough to make a practical difference. The listening-to-music diet won't be effective.
As you suggest in your question, it is very plausible that listening to music may increase you heart rate, cause you to start tapping your foot or jigging your leg which in turn will burn calories. People often find that listening to music has the psychological effect of allowing them to do boring repetitive tasks (including "exercise" or running/jogging) more intensively or for longer, so there can be valid indirect effects.