Some things should be left to the professionals(half joking half serious).

Electrocution is a motherfucker.

Did he ground the switch box or the fixture housing?

I don't see that there is anything he could have done wrong that would cause that behavior. The only wrong thing he could do that would still result in the light shining is to reverse the hot and neutral (assuming all he did was swap the fan for the light). That's a very, very bad thing to do, but wouldn't cause this in an otherwise correct circuit. I don't think anyways. I'd have to resort to drawing pictures to figure it out.

This is probably a good thing because it means he discovered a potentially dangerous problem in the previous circuit. There's lots of ways you can wire up two switches a light and a GFI socket that appear to work but are actually dangerous mistakes of varying degrees.

ed: My suspicion is that the whole circuit is wired to a different breaker on each end, which is a mistake I've seen more than once in the few years I did this stuff and a giant pain in the ass, not to mention dangerous... though he would be lucky not to discover that while changing the light, unless he just turned off all the breakers or something.

*promote

Promoting this video and sending it back into the queue for one more try; last queued Thursday, February 26th, 2015 2:09pm PST - promote requested by blackfox42.

Also I documented it, so if things burn down, you can see I am doing unlicensed electrical work and caused the destruction of lives and property.

Since he mentioned that he replaced an old fan with a light, those run on 3 wires (white, black, red) plus ground (earth)- in order to turn the light and fan on/off independently. He must have reconnected those incorrectly in the ceiling. Now the hair dryer acting as a switch means that outlet the hair dryer plugs into is wired incorrectly, and is not a laughing matter.
Yes, he should call an electrician and avoid killing anyone.

This just as easily could have ended up a Liveleak video, with him dropping to the floor with sparks and a zzzzzzt sound effect. With her annoying giggling turning into annoying shrieks.

I do all my own electrical work. I have a degree in Electronics which certainly helps with understanding though the classic HOME WIRING books were a must too. I found serious problems with the house we bought. Someone had bonded neutral and ground in a subpanel and we had voltage on every metal surface in the house.

Electrical Engineer here. Still haven't figured this one out yet.

*engineering.
Someone's coupled something in serial - the light is (also) coupled in serial with maybe several other outlets - causing even higher voltages, and causing the light to go on when the hair dryer circuit connects.

Adding video to channels (Engineering) - requested by Zawash.

this could also be a "shared neutral" situation where more than one circuit shares a neutral return.