How can I significantly increase the temperature in a relatively medium sized wooden box effectively to 300∘ 𝐹? The box is 3.5 feet in height, 2.2 feet in width, and 9 inches in length. Also I put insulators (like styrofoam) around the box with lots of tape to make sure no heat has escaped. I've inserted a hole where a hair dryer goes into the box so that it can generate heat (around 100 degrees in heat). I've also tried decreasing the size of the box, any ideas or tools that can help generate more heat? Preferably 100−200 degrees more to reach 300∘ 𝐹?
Do you know the burning temperature of the wood you intend to use? According to a quick search on the 'net, you're in trouble at about 450° F. Extended temperature at that level will ignite the wood.
Your insulation should be inside your enclosure and the enclosure should be non-combustible.
A hot air gun blowing into a well insulated enclosure will likely reach your objective temperature.
The code for the temperature symbol - ° - is alt-0176 unless that's what you used and the italics code has shifted it downward.
First of all: DON'T TAKE WHAT IF? AS ADVICE. Anything that involves hair dryers in boxes will start fires over most of North America.
According to Sciencing, "wood" will start to burn at 570° F or 300° C. Fire Engineering magazine mostly clarifies this, saying wood at this range of temperatures will "gradually char and usually ignite after several hours", though at the lower temperatures of 450-500° F or 232-260° C. (I would read the specifics, but then I'd have to subscribe to the magazine.) (Also, there must be different ignition temperatures for different kinds of wood, but none of the sources specified what kind they were using.)
Long story short, your box will catch fire after a while, and will char significantly before that. That's not a good side effect of "dehumidification."
It is a cool idea to use hair dryers, though. They work by heating air and blowing it over the thing that needs to be dried (most often hair). Hot air speeds evaporation by both heating the water and having greater humid capacity. Moving air will also speed evaporation (which is why you feel cold after stepping out of the pool on a windy day).
Rather than increasing the temperature of the air to the ignition point of wood, why don't you set the hair dryer at a cooler temperature, make sure the box is well ventilated, check for any hot spots (like right in front of the dryer if you face it towards the one of the walls), and give it a few hours?
If your goal is simply to heat a box to a really high temperature, though, I'd advise you to use a different material.