The door has been used mainly in the film as a transition between the life of the mafia and the life of a normal person. All of the quiet and criminal acts are committed inside and all the normal everyday life events are committed outside. For example when Tom Hagen left that wonderful $600,000 horse head on Woltz feet. That was committed behind closed doors. Every criminal act that is committed in the godfather is done behind closed doors. That opposed to the nice scenes like the wedding are all outside. As stated above all the scenes involving the criminal empire of the mafia are done inside and all the normal life scenes are done outside. Coincidence? I think not.
Doors mean a lot of things. In the Godfather, they represent the transition between the mafia and normal life. If any other meanings of doors are represented in the Godfather, I would say they would be in the future scenes.