The VS10xx family uses a virtual ground for its audio output. If you connect the virtual ground (1.6V) to real ground (0V), an internal short will be created that WILL damage the chip.
You CAN connect the audio output (headphone jack) to a grounded audio input IF you do it through some inline protection circuitry. This application note has the details, and one of our customers put together a nice diagram of what you need to add on their website. Once you have this circuitry in place, you can safely connect the LilyPad MP3 Player to any amplifier, whether the LilyPad is battery powered or not. (And yes, this goes for the MP3 Shield, MP3 Trigger, and VS10XX breakout boards as well).
I hope this helps, if you have other questions let us know and good luck with your installation!