...the description text of the box object should be rendered dynamically on the sides of the cube.

Unfortunately that is not possible. You can display the description but that is only on top and only in the 2D view. That display will change when you change the description. You probably want to see it in the 3D view so that is not an option anyway. Even then, it is one description for the full object, not for/on each side.
There is an old wish to be able to attach a label/textbox to an object. And in such a way that it moves along with the object when you move it. Something like that could be a (partial) option for you (if you could attach multiple labels). But as said, it's an open wish, not an available feature.