iPhone Headphone Jack - Accessory & Headphone Use
The HiJack system uses the headphone jack to provide power to and exchange data with your accessory.  As a result, the headphone jack is not available to play music at the same time.

With HiJack, the iPhone communicates with the accessory at 8.82 kbps.  I suppose that it would be possible to write your iPhone app in such a way as to multiplex some audio along with your accessory data, then have a Digital-to-Analog converter on your accessory process the audio data and send it to a speaker or a headphone jack, but you won't have enough bandwidth to get a decent sound quality for music.

HiJack is also limited in the amount of power that is provided through the jack (7.4 mW).  It would be difficult to power your circuitry and a headphone without external power.