I think I would only run it for 5 or 10 minutes at a time, at cold starts. When my wife dries her hair it definitely runs for longer than that... To power the 1000 watts of heat you would need a mighty stout inverter...
I have been thinking the exact same thing you have, (a hairdryer!!!) except I was going to turn the temp. on the hair dryer to "warm" for a warm air intake. Also, i was thinking of removing the fan from it and leaving the heating coil inside so it would just be considered a heater.
On EFI equipped car there is an intake temp sensor, this sensor helps determine how much fuel to inject and how much timing the air can handle. Higher temperatures mean the air can handle less fuel before it goes rich, conversely the cooler the air the more fuel it can support. Since we here are not so concerned about peak power and high MPH, high intake temps are a good thing. My car for instance has manifold pressure based fuel mapping but the IAT (intake air temp sensor) determines the offset on a given map. I run a hot air intake (as seen in my garage) my intake temps run between 130* and 140* f compaired to the 78*f without, my fuel usage went from 1.5 gallons per hour (GPH) to just 1.09GPH at the same throttle setting (cruise 65 mph) this netted a huge gain in cruise milage. Again we aren't looking for low quarter mile times just good MPG's.
phil hit the nail on the head. less power but more FE. the trade off is there and has always been there. I have heard that it doesn't work so well on all vehicles but I personally saw about 10% gain out of mine too. taking the heat off of the header (exhaust manifold).
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Cool air is good for making more power but warm air is better for mileage because it helps vaporize the fuel better. Unvaporized(liquid) fuel doesn't burn. Cool air would not allow as much vaporization.
Less O2 = less gas.
So hot air effectively makes the motor smaller...