I think you mean shut the door, not shoot the door. Both phrases mean the same thing. "Could you shut the door", means the same as "Could you close the door". For more tips on language learning please visit http://language.today.com/

For all practical purposes they are the same, although I think many people would say 'close' is a more polite word than 'shut'. Why? I don't know, maybe because of association with the impolite phrase "Shut up." The precise difference is that closing is about the action of closing, like a door may be in the process of closing but not closed yet. A light switch, by comparison, moves so quickly that it is either on or off. We probably don't talk about the time it is changing. So we say 'shut off the light' because 'shut' is about the finished state of being closed. (An electrical engineer might talk about closing an electrical circuit, but that's technical language people don't normally use.) 'closed up' and 'shut up' mean the same, for instance a store that is closed up or shut up.

if you say "shoot the door" to me it means something like 'shoot the door WITH A GUN' if you say "shut the door" it is something more than "close". like it is closed tight or closed with more force or closed with a bang

Both phrases seem equal. Although words shut and close are similar in meaning, one word can not always replace the other. "To shut" would be used for doors/gates and so on suggesting a mechanical/literal process. "to close" has more wide and abstract meaning: f.ex. closed question, etc. Further, the use of shut is more common to indicate noise. Hence, shut is usually follows words such as slammed and snapped because of the convey sound. E.x. the door snapped shut. "Close" indicates more slow process. Besides, shut may seem slightly impolite.