The phrases were you born in a sawmill? and were you born in a barn?, and their variants:
– express mild remonstrance towards a person who has left a door open, exposing others to a draft;
– indicate that a person is behaving in a rude or uncouth manner.

These are, in chronological order, the earliest occurrences of each of these phrases that I have found:

 

WERE YOU BORN IN A SAWMILL?

 

1-: From The Weekly North Iowa Times (McGregor, Iowa, USA) of Wednesday 12^th December 1860:

The Door.
Do you know anybody who never shuts the door—a fellow who was born in a sawmill and raised on a raft? Tell that chap “it is Winter; wood is high; it takes a hot stove to make the family or office comfortable,” and then invite him (respectfully at first) ‘to shut the door’ and quickly too.

2-: From Local Brevities, published in The Morning Republican (Scranton, Pennsylvania, USA) of Wednesday 18^th February 1874:

—The following gentle hint appears on the door of a certain establishment, not a thousand miles distant: “Was you borne in a saw-mill? if not, shut the door.”

3-: From Town Talk, published in the Providence Evening Press (Providence, Rhode Island, USA) of Monday 20^th December 1875:

—People who were “born in a sawmill” have no business in such weather as this to live among civilized people who are not habituated to the draft from the door or window.

 

WERE YOU BORN IN A BARN?

 

1-: From one of the unconnected paragraphs making up Council Jottings, published in The Cleveland Leader (Cleveland, Ohio, USA) of Wednesday 19^th July 1876:

That young man who frequented the gallery with his hat on was probably born in a barn and brought up on the frontier.

2-: From Out of Fate’s Labyrinth, a short story by Mary Reed Cromwell, published in The Cincinnati Daily Star (Cincinnati, Ohio, USA) of Monday 7^th June 1880:

The earliest train down took him to the city the next morning—a sharp, crisp winter day, with plenty of snow in the air, and a biting northeast wind blowing; a gloomy, depressing sort of a morning, when travelers were not in the best of humor; and heedless people persisted in leaving depot cars ajar—notably one tall, angular woman, with a white nubia tied over her bonnet, that made her face ghastly in contrast, and dressed more with regard to comfort than appearance.
“I’ll thank you to keep that door shut, madame,” Morgan said, at last, exasperated by her repeated journeys from the waiting-room of the ferry-house to the pier outside, and the invariable neglect of closing the door after her.
She turned sharply on him.
“You would, would you? Well, when you want it shut, shut it yourself!”
And she bestowed a malicious laugh on him that brought the angry blood to his cheeks, and that made him lose his temper completely.
“Of course, if you was born in a barn, all right, ma’am. Some folks know enough to shut the doors, and some don’t—that’s all!”

3-: From Shop-Door Warnings, published in the Plain Dealer (Cleveland, Ohio, USA) of Saturday 3^rd September 1881—reprinted from the New York News:

These store placards are not all of this grim and threatening sort, though. There is the one which invites the visitor to shut the door. This invitation assumes various forms. In one a man looks out of a window with a blunderbuss in his hand, roaring: “Shut the door!” In another a simple question is propounded: “Were you born in a barn?” Another still informs you: “We like our doors shut.” And still another asks: “Did you find the door open?”

 

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Two synonymous phrases, were you born in a tent? and were you born in a field?, occur for example:

– In Two-minute chats. With busy men. On matters of passing interest. (By our peripatetic reporters.), published in The Australian Star (Sydney, New South Wales, Australia) of Thursday 14^th June 1906:

WHICH IS IT?
Is the present race of Australians entirely selfish, are they merely careless, or lazy, or is it simply perverseness of human nature?
These questions, apropos the “door” problem, were qut [misprint for ‘put’] by a “Star” reporter to his neighbour in a railway carriage. It was a cold, wet night, yet not one person out of fifty, either entering the carriage or leaving it, would shut the door behind him.
“I don’t know,” said the person addressed. “That same thing has often puzzled me. If women were the worst offenders you could, perhaps, understand it on the ground that they are brought up to expect things of this kind done for them by the gallant sterner sex. But they are not. It seems to be pure ‘cussedness,’ as the Yankees say, very often. A man will come into the carriage from the end next the engine, and someone near at hand will shut the door after him. He will take a seat three or four seats down, and then someone will open the door, walk out, and leave it open. When the train starts the cold blast will come in and strike the man who, although he had left the door open when he came in himself, will go and slam it, and make some sarcastic remark about people being born in a tent, or words to that effect.”

– In Temper-atures now 40 below affable, by H. de Winton Wigley, published in the News Chronicle (London, England) of Wednesday 12^th February 1936:

People are saying that the searching east wind is spreading a wave of irritability over the British Isles.
So that’s why nobody likes me these days! And all because this icy blast suits me down to the ground, fills me full of beans and pep and energy, and makes me thump people on the back.
I like this weather. I decide for three-quarters of an hour that I must get up immediately, but once up, things move rapidly.
I come in with red ears and rubicund nose, glowing and hearty, to find people crouching miserably over the fire sniffing and nursing chilblains.
COLD RECEPTION
I remark that “this is the weather to put life into one,” and they glower at my glow and say for the love of Mike will I close that door and was I born in a field.

The following is from The Sydney Morning Herald (Sydney, New South Wales, Australia) of Tuesday 16^th November 1937:

CLOSE THAT DOOR.
Almost as long as there have been houses, there have been doors; and as long as there have been doors there have been people to leave them open.
It is not surprising that various sayings have sprung up to be directed at these careless folk when they are caught in the act.
In Australia it is customary merely to inquire: “Aren’t there any doors where you come from?” But in other countries different phrases have come into being, some of them not quite clear.
In Canada it is usual to say, “Were you born in a sawmill?”
In England failure to close a door earns the request: “Were you born in a field with the gates open?”
Irish people cry, “Gorey!” when a visitor omits to shut the door after him. Here there is a reasonable explanation. During the revolts of the late eighteenth century *, the people of the town of Gorey tore down their doors and used them as barricades. When the town was finally subdued, the commanding officer ordered the doors to be burned, thus punishing the rebels and branding their homes until further doors could be made.

* This refers to the uprising against British rule in Ireland, which took place from May to October 1798. Gorey is a town in County Wexford, Ireland.