Archaeologically speaking, we see doors being used in structures from around the Neolithic onwards. Of course, the date of the Neolithic varies hugely by region. So in my neck of the woods (northern Scotland) we see pivot points for doors appearing around the early 4th millennium BCE (at sites such as Skara Brae, Ness of Brodgar, Barnhouse etc.). These doors were typically stone slabs that had been shaped to pivot upon a socket stone, and could then be barred by a piece of timber slotted into the stone door jambs (see Prof. Colin Richards 1993 PhD thesis).

Further afield, we see evidence of doors in early urban societies that predate (just) the British Neolithic, including Sumerian architecture (4th millennium BCE) - although other early urban (or proto-urban) societies (such as that of Çatalhöyük) did not have external doors.

However, if you specifically mean timber doors with actual hinges (rather than a solid slab with pivot points) then I think the earliest examples are found in Ancient Egypt (see Clarke and Engelbach 1930) - dating to around the middle of the 3rd millennium BCE.

Is it appropriate to piggyback on this question and ask if anyone knows the history of doors in Japan? The word for door is literally "dooah", its written in katakana which means it is a word borrowed from western culture, which always struck me as odd. Did they not have hinged doors prior to contact with the west?