Do some headsets demand more power than others? Since I started using a new set, the volume has been lower, but the battery has been draining much faster.
In a word? Yep.
What factors might cause a headset to drain power?
Others will know better, but it’s the construction and electronics of the phones. Mini-amps (such as those by FiiO) can be good to handle this.
Better quality headphones/earphones often (but not always) have higher impedance (which puts more drain on the battery)and lower sensitivity (which means you have to turn up the volume of the MP3 player to get an equivalent loudness of sound).
There are some high quality headphones/earphones which have low impedance (quoted in Ohms, look for <100) and high sensitivity (quoted as dBs SPL/mW, look for >100).