With reference to the traditional use of white coats by medical personnel, the humorous phrase the men in (the) white coats denotes psychiatrists or psychiatric workers.

This phrase is often used in the stereotypical image of a mentally-disordered person being borne away to a psychiatric hospital by psychiatric workers.
—Cf. also loony doctor and loony bin.

These are, in chronological order, the earliest occurrences of the phrase the men in (the) white coats that I have found:

1-: From the Quincy Patriot Ledger (Quincy, Massachusetts) of Saturday 9^th May 1936:

Should a man walk up to you and offer you $500 to remove a valuable article from his store, you’d undoubtedly blink twice, clap your hat down hard on your head, and run to the nearest phone to call for the men in the white coats.

2-: From the sports column Thisa and Thata, by Lin Raymond, published in the Quincy Patriot Ledger (Quincy, Massachusetts) of Friday 14^th January 1938:

The difference between a fair season and a good season for the North Quincy High school basketball team, can be indirectly traced to the work of a vandal, who must have had a screw loose somewhere. Yes, indeedly.
The announcement that Capt. BILL “TIMER” RYAN would not be ready for action until next week, brings to light more than ever before the harm that one missle [misprint for ‘missile’] had in wrecking what loomed as a fast team.
[…]
It’s too bad that other people have to suffer for the wrong doings of certain people who have been evading the men in white coats!

3-: From New Formula for Bowl Bid: Just Sign ‘Expert’ for $4000 and Get Unstoppable Attack, by Bill Cunningham, published in The Boston Herald (Boston, Massachusetts) of Tuesday 29^th July 1941—the author first quotes, then comments on, a letter that he has received from “one of the greatest punters in U. of Michigan football history”:

“I have now developed a system of attack which I do not believe can be stopped […].
“[…]
“What I would like to get out of it is a contract as Consulting Coach for one year at a salary of $4000, plus 10 per cent. of any amount received for post-season games. […]”
[…]
The quick reaction, of course, is to look around for the men in the white coats.

4-: From Sweet and Low-Down: Any Day Now a List of 10 Best Vocalists, Minus Bob Eberle, Will Be Forthcoming, by George Frazier, published in The Boston Herald (Boston, Massachusetts) of Sunday 1^st March 1942:

So if I don’t like Bing Crosby’s ten favorite vocalists, why don’t I pick ten of my own? Generally speaking, that was the somewhat infuriated tenor of this week’s mail. In reply thereto may I state that I intend to […]. One thing is certain: I shall not pick Bob Eberle […].
On this question of Eberle, most people think that he is pretty wonderful and that I am crazy. Maybe so. But I still don’t like him and him you can have […]. I think it might be rather interesting to find out just who are your ten favorite popular vocalists, both male and female. If you think so too, I wish you’d send me your selections. […] I’ll get the composite result of all your votes. That result will appear in this space in, say, three weeks. I’ll run my own personal selection alongside it and then wait here patiently for the men in the white coats to come and take me away.

5-: From the column Facts and Whimsy, by William G. Schofield, published in the Boston Traveler (Boston, Massachusetts) of Friday 13^th March 1942:

The little men in the white coats scampered in and took away Mrs. Doakes today.
Up to the moment they were called, she’d been playing a game called “What’s the difference?” By the time they clambered in her window, she was fair game herself, for anybody with a butterfly net.
Mrs. Doakes was the die-hard isolationist leader of the neighborhood sewing circle. After the nation got in the war, she amended her total hands-off arguments with an attitude that claimed it didn’t make any great difference to us what the Japs did in faraway places like Batavia and Manila and if we had any brains we’d call the navy home and tuck in our shirts and sit down behind a North American defense line that couldn’t be cracked.
Having taken her stand, Mrs. Doakes then offered to prove to the rest of the girls that anyone could get along without a thought for the lands beyond the Pacific. She said, said she, that she’d just as soon go through a whole day without using any of the products from that vague part of the world, just to prove that we could get along quite merrily even if we never had any more trade with those markets.
Since none of the girls liked Mrs. Doakes anyhow, they told her to go ahead and try. And one of them was thoughtful enough to call up the place where the little men stay and tell them to put on their white coats and get ready for a frolic.
Well, the Doakes dame found herself dunked in a stew right at the start.
First she had to shed her silk stockings and her silk undies, and that put her in a sour mood to begin with. She didn’t see why we couldn’t get a happy pair of silk worms and start them housekeeping over here instead of letting them all go to waste in Japan.
Woman-like, she figured a fresh shampoo and hair-do would perk up quite nicely, so off she trotted to the wash bowl to treat herself to the works.
But she had to cancel that order when she discovered there was quinine in her hair tonic. And just as she was about to take a quick drink for moral support, she learned there was also quinine in her mixer—and quinine in the medicine cabinet and quinine on the pantry shelf.
She couldn’t take a shower, because there was tung oil in the bath curtain—tung oil straight from China, to keep the water from leaking through and dripping on the floor.
The same sort of tung oil, she discovered, had a definite effect on the waterproof quality of her raincoat and on the oil-cloth that covered the closet shelf and on the varnish on the floor and on the paint that made her kitchen so cheery.
She couldn’t have a cup of tea, or even a cup of cocoa. She couldn’t eat tapioca or even make herself a rice pudding. And when she finally got around to whipping up some sort of a meal for the afternoon, she found she couldn’t use black pepper or white pepper or nutmeg or cloves to season the food she took from the canned shelf—and as a matter of fact, she couldn’t even take the food down to begin with because it was all wrapped up in tin.
Along about this point, Mrs. Doakes decided she’d had enough for one session and hied herself to the studio couch for a few minutes of rest and recovery.
That business ended as soon as it started. For no sooner had she plunked herself down on the cushions than she had to plunk herself up again to get away from the kapok stuffing. And no matter what piece of upholstered furniture she tried to sit in, she found the same kapok problem there ahead of her.
Somewhere along in here, Mrs. Doakes really cracked. She went screaming around the house, yelling threats at the Nazis and the Japs and pleading with somebody or anybody to let her get her hands on a rope so she could string Hirohito from the chandelier.
Only then did she learn that the best rope, for hanging and all other purposes, is the Manila hemp that comes from the abaca fibre in Davao. And Davao, she was reminded, had gone out of the rope business except for Japanese consumption.
That was when the little men in the white coats clambered through the window.
They found her rolling on the floor, laughing and moaning and mumbling something that sounded like:
“We’ll show ’em! We’ll show ’em! The dirty little ——!”
They never did catch that last word of hers. It was a new one, even for the tough little men.

6-: From an article about turfmen, by Whitney Martin, of Wide World, published in several U.S. newspapers on Thursday 11^th June 1942—for example in The San Diego Union (San Diego, California):

If a guy wants to stand on a street corner and hand out silver dollars that’s his business, although a wagon probably would be backing up before long and a couple of men in white coats would say gently: “Come on Napoleon. Josephine has supper ready.”