My informant first heard this phrase when he was a young boy, originally as “Hotter than a fox in a box in a forest fire” but over time, began saying the phrase without the fox included. He says it is just quicker to say.
He is from Sacramento, California and grew up hunting and fishing outside the city and in the mountains near Lake Tahoe. He is not quite sure if he heard the phrase from his father, grandfather, or uncle, but thinks he may have learned it while on a hunting trip with them.
The phrase refers to a person feeling very hot temperature-wise, to the point often of sweating or feeling uncomfortable. My informant says he uses the phrase most when in a hot car, or in a crowded, stuffy room.
A forest fire is known for being dangerous due to the fast pace with which the fire spreads. With loads of trees to burn as fuel, and abundant oxygen, the fire is very hard to contain. Forest fires most often occur when it has been very hot and very dry, usually during the summer. The heat of the fire, plus the heat of the summer air, makes an intensely hot combination.
Now a fox, is a forest dweller, known for its red colored fur, and red is the color of heat. People once hunted fox for their fur, which was used for warmth, so we know a fox is already well insulated from cold. In a box, surrounded by fire, that is basically raging uncontrollably, in an already hot temperature, we can safely assume the fox would be burning alive. So the simile maintains that a person is extremely hot, if they are hotter than a fox in a box in a forest fire!