While not sure, I think it depends on how you're routing your sound output. If your sound it going out through the HDMI, but your TV is off, you're out of luck. Others should confirm this. I have an aTV3 in transit. I'll know when it arrives.
If your audio plays through a receiver, which the atv is plugged into then turning the tv off shouldn't affect it. When I am not using the tv I keep the receiver on with the atv input selected so I can play music whenever I want without having to turn it on. I actually programmed my harmony remote so when I turn the tv off it leaves the receiver on and switches to the atv input. Then the speakers are ready to stream music whenever we want. works really well.