I'd love to see that actually... Most hair driers are very power hungry. 2kW is not uncommon (at least here where standard voltage is 240V). Achieving substantial amount of drying from a mobile phone battery would be quite a feat of engineering.
Make a small rechargable hair drier for camping/use where outlets are in short supply, and you've got yourself a saleable product, even it it only gives about 5 min of hot air.
If battery powered, it would be a very expensive device and require a big heavy battery.
Power tool batteries, eg a dewalt 18v drill, allow a max discharge current of around 10 amps. So for a 600 watt hair dryer, you're going to need 3 power tool batteries! Imagine the cost/weight of that!
To make it work without the high cost/weight, you're going to have to blow mostly cold air and hope the user is okay with that.
