"Open the door for me."
Translation:Otwórz mi drzwi.
I would distinguish two slightly different situations here.
1) You are carrying a heavy box and can't use your hands, but want to exit the house. You would say to somebody "Prosze, otwórz mi drzwi" - here, a fact that you need the door to be opened for you is emphasized.
2) You are sitting on a coach, it is very hot in the room and you are too lazy to stand up. You ask someone else "Prosze, otwórz drzwi dla mnie" (or even "za mnie", "zamiast mnie" = instead if me) - here, you emphasize the fact that you are a narcissistic bastard :D
It seems the dative case "mi" is used here because the meaning is "for me". And the alternative translation of "dla mnie" doesn't use dative there because you are already explicitly using the word "dla", so don't need to convey that meaning with the dative case (plus "dla" demands genitive). Am I on the right track, or am I thinking about it wrongly?
Seems right to me. "Otwórz mi drzwi" would be like "Open me the door", but I don't know if such a sentence would be correct English. And "Otwórz drzwi dla mnie" is literally "for me", so this one's pretty clear. Anyway, "dla" takes Genitive, so it has to be "mnie".
EDIT: Apparently "Open me the door" is okay, although uncommon.