MP3 Player, does volume affect power use?
Yes, most likely higher volume means higher power.  This is particularly true for devices with limited power so that were therefore designed to conserve power to a reasonable extent.  A portable MP3 player running on batteries would be such a device.

Most likely, the MP3 player has a class D audio amp chip in it to drive the headphones.  This is the most efficient way available to produce a power audio output signal.  These things can be quite efficient.  For a perfect class D amp, all the input power would go to drive the output.  The output power is the square of the output voltage divided by the load impedance.  If the output voltage doubles, the output power goes up by a factor of 4.

In reality you won't see a factor of 4 power draw increase from the batteries when the output voltage is doubled because there are losses in the system and some fixed bias currents for various circuits.  However, there should still be a significant effect.