How do you remove insulation from headphone wires (these tiny) to prepare it for soldering?
On smaller, thin wires, simply applying heat from the iron will delaminate (ie, remove insulation from) the wire. I've done this (by accident - creates interesting situations in a wiring harness if you are lame like me and try to shrink heat-shrink tubing with an iron - don't do it!).
You might be able to use a chemical agent to strip the insulation too.
You could also try a razor blade, but you need ninja-class hand-steadying abilities.
I just did some soldering with small wires and noticed that if you heat up the cable, insulation will shrink back from the heated part. I usually use sharp scissors to remove a bit of insulation at the start of the cable and then just start soldering on the small exposed part. If I'm quick enough, insulation will shrink just enough to not be a problem.
If you need to remove insulation from the middle of the cable, I have no idea how to help you.
EDIT: I just did some soldering to a middle of a cable and I've noticed that on thin audio cables it is possible to just solder directly over the insulation. It requires a bit more time and heat, but insulation will break and shrink from the heated area leaving a piece of copper conductor exposed.