A/C off windows open, same difference. [?]
Hello and welcome to the RTA forums !
Obviously the A/C can cause some loss in power, but the radio would not make any difference IMO. One of the great debates about having the A/C turned off for more power and better mpg consumption, is that many believe by switching off the A/C and then having the windows open instead, causes increased drag by air buffeting through the car. The 'drag' then has the same effect as reduced power output, and will use more gas. This is most likely what your friend has heard, that by turning the A/C off and opening the windows means there is no 'gain'.
Personally I think it is negligible either way, particularly on more modern vehicles.