When discussing the uses of aluminum foil, the question of which side is up will be raised. If you're going to use it to store heat, the shiny side must face the sun. For other uses, it doesn't matter which side is up.

The shiny side is slightly better reflector of heat. Face that side of the foil in the appropriate direction. To keep things cold, put the shiny side on the outside -- that will reflect incoming heat. To keep things warm, face the shiny side inward toward the hot food -- to reflect the heat that is trying to escape back into the food.

Note that people often bake potatoes with the shiny side out -- that is because it makes for a better (prettier) presentation. In reality, baking them that way reflects the incoming heat on the outside, slightly slowing the cooking process.

Of course, in reality, which side of the foil faces where makes little difference. It does make for lively conversation, though. There is no difference in which side you use. The difference in the surface of aluminum foil is a result of the manufacturing process and has no bearing on heat retention or reflection.

But the target "hamburger-in-foil-on-the-grill" is, in practice, very complicated. The charcoal is radiating heat to the burger from below (which is highly variable depending upon the surface temperature of the coals. On top the burger-foil is losing heat energy to the surroundings, but it is virtually impossible to assign an "average" temperature to the surroundings. To make matters worse, the aluminum foil is "crinkled" so that the reflecting surface is not uniform, and on the "dull" side the reflectance is not even specular, i.e. there is a range of angles of reflection for a specified angle of incidence, and that "range" depends upon the angle of reflectance.