Question about pressurization and introducing air into a small chamber
Hair dryers are typically similar power compared to heat guns, maybe higher, though the air flow is usually a lot higher. Somewhere in the 1500-1800W range for North American ones, which is about as much juice as you can get out of a standard 120VAC outlet. 

If you block off the air flow from either one, the internal thermal protection will (should) trip to prevent the case from melting (in either case). 

So, yes you need to allow the air to escape, but if it is well insulated and the heat is not controlled the air temperature will likely eventually rise to a high enough temperature to melt the styrofoam etc. That will depend on the size of the chamber and the heat capacity of whatever is in it. To keep that from happening you will need to use the hair dryer and keep the air flowing as freely as possible so it does not get too hot. It may still get too hot if you can't keep the air flowing freely enough.