Unless the space heater has the potential of getting the inside of the box to over 451 degrees, and there's nothing in the box that can combust at a lower temperature than that, I don't know that it is a fire hazard. A space heater than can generate even 150 degrees would be too hazardous to use around people, since you can get a third degree burn from just two seconds of exposure to that heat. The box obviously isn't sealed, so heat can certainly escape from it. I would say that lining the box with foil might make it more of a hazard, since foil is going to reflect heat back into the box. Whether or not that would be enough to get the temperature up to 451 degrees would depend on the heat source. The small one I have (which looks to be about the same dimensions as Richarddacats's) probably couldn't get that box up to more than 80 degrees. There's always the chance that the cardboard would somehow come in contact with the heating coil in the space heater, but that's kind of a remote chance.
You could always jam a digital thermometer through the side of the box, if you wanted to make 100 percent sure.