After playing with different audio session properties, I found that -16981 error takes place when kAudioSessionProperty_OverrideCategoryMixWithOthers is enabled (TRUE). As soon, as I set it to '0', AudioQueueStart() executes successfully. So, before starting the audio session try:
UInt32 allowMixing = 0;
status = AudioSessionSetProperty (
kAudioSessionProperty_OverrideCategoryMixWithOthers,
sizeof (allowMixing),
&allowMixing);
Clearly, this is behavior change in iOS 7. As it was mentioned before, the documentation does not list -16981 error code.