|MadSci Network: Physics|
Although this question has been addressed before in the MSN (see below for references), I will elaborate on it and give maybe some new insights.
First of all, what you describe cannot work because the reflection of light by a mirror is not a dissipation-free process, i.e. light loses some intensity with every reflection. As the speed of light is so incredibly large, there are very many reflections per second taking place inside the box. So after some ``flash'' of light has entered the box and the lid is shut, dissipation and the large number of reflections ensure that all the energy gets transformed into heat in the course of a very tiny amount of time, so small that the eye cannot resolve it.
But let us consider the case of a very large box; it could, for example, have an edge which is as long as the distance between the earth and the moon. Then it would take approximately 1.3 seconds for a flash of light to get from one end of the box to the other, and there would be just a few reflections per second taking place. In such a case, it would take much longer for the light to cease after shutting the lid. This situation is analogous to the sound of echoes in large and small rooms, respectively. In a small room, you do not hear any echo because the sound waves get reflected so soon after production that your ear cannot dustinguish them from the original sound, and after a few reflections they are converted to heat. But in a large room like e.g. a hangar or something, the echo is very pronounced.
You see that the desired effect depends crucially on two factors:
For reference, I am providing links to two other questions which have been submitted to the MSN and which are closely related to your question: What happens to light? and Where does light go?
Hope that helps,
PS: After telling my wife about this problem, she raised the interesting question how a refrigerator manages to keep the light inside -- because every time you open it, there's light coming out... ;-)
Try the links in the MadSci Library for more information on Physics.