The phrase the lunatics are running the asylum, and its variants, mean: those in charge of an organisation, project or initiative lack the fundamental qualities needed to fulfil their responsibilities.

This phrase was popularised by the U.S. film producer Richard A. Rowland (1880-1947), who used it when reacting in 1919 to the establishment of the United Artists Corporation, which allowed actors to control their own interests independently of commercial studios.—Cf., below, quotations 5, 6, 7, 8 & 9.

Apparently, the phrase the lunatics are running the asylum did not originally allude to The System of Doctor Tarr and Professor Fether (1845), a short story in which the U.S. short-story writer, poet and critic Edgar Allan Poe (1809-1849) described such an event.

Incidentally, similar events have occasionally happened. The following, for example, is from the Belfast Telegraph (Belfast, Antrim, Northern Ireland) of Friday 21^st February 1919:

MONAGHAN ASYLUM STRIKE.

Major NEWMAN (U. Enfield) asked the Chief Secretary in the Commons whether he was aware that the medical officer and staff of the Monaghan Lunatic Asylum, with the aid of the inmates seized, at the beginning of this month, the asylum buildings, and established a local soviet or committee, consisting of the staff and inmates, to run the asylum; whether the local authorities were unable to effect an entry to put an end to the committee’s rule, and will he say what is the present situation?

The earliest occurrences that I have found of the phrase the lunatics are running the asylum and variants are as follows, in chronological order:

1-: From the account of the meeting of the Town Council that was held at Kingston upon Hull, Yorkshire, on Friday 22^nd January 1886, published in the Hull News (Kingston upon Hull, Yorkshire, England) of Saturday 23^rd January 1886 [page 6, column 1]:

The Mayor […] called upon the Town Clerk to read the resolution.
The Town Clerk said it was as follows:—
That, having regard […] to the representation of the Boards of Guardians of the joint Unions that […] the six Workhouses are overcrowded, the Foundling Hospital is inadequate in size, and the Lunatic Asylum needs to be enlarged, while the Poor-rate for the ensuing year will be 15s in the £.
[…]
Mr Alderman Holderness said […] they knew what he intended to say, and how it was that they now had six workhouses, and then only had two, and why the gaol had been doubled in size, and why the lunatics were overrunning the asylum [laughter] and why the rates were 15s in the £. As to the Foundling Hospital, he did not care so much about that.

2-: From the account of the quarterly meeting of the Flintshire Police Committee that was held on Thursday 24^th August 1893 at Mold, Flintshire, published in The Wrexham Advertiser, and North Wales News (Wrexham, Denbighshire, Wales) of Saturday 26^th August 1893 [page 5, column 7]:

Mr Llewelyn Jones was glad to hear the statement, but he wished to know what about the other, for it was obvious that a man who got drunk was incapable of maintaining order in his house. It was like giving a lunatic the charge of a lunatic asylum.

3-: From The Jefferson County Tribune (Oskaloosa, Kansas, USA) of Friday 6^th April 1900 [page 5, column 5]:

Are the Inmates in Charge at the Asylum.

James Morrow, John Gibbs, Randolf Moss, John Ramsey, E. D. King and Dr. McCreight constituted the jury in the Christopher Hanson insanity case. Hanson had been to the asylum before and Judge Dick knew it to be unnecessary to incur the expense of a second hearing and asked that it be dispensed with. The authorities at the asylum telephoned that a new verdict must accompany the subject and one was obtained and Sheriff Newell went to Topeka with Hanson Tuesday. Then a letter came from the asylum stating that a new verdict was not necessary—but the expense was already incurred. The change of mind cost this county about $40. Do the inmates have charge up there?

4-: From the transcript of the address that the Rev. Dr. E. L. House, of Westminster Congregational Church, delivered during the Memorial Day parade that took place at Spokane, Washington, on Saturday 30^th May 1908, published in The Spokesman-Review (Spokane, Washington, USA) of Sunday 31^st May 1908 [page 9, column 1]:

“The time has come when we should restrict immigration. The time has come when the man or woman who has no other object in view than to incite to treason ought to be sent back to the shores that gave him birth.
“It is not a question as to whether you are born in America, but rather if America has been born in you. If we must have asylums for the criminal and the insane let us not put lunatics in charge of those asylums.”

5-: From Photoplay Magazine (Chicago, Illinois, USA) of May 1919 [page 73, column 1]:

Mr. Rowland’s Rejoinder.
The picture magnate who has a sense of humor generally refers to his own estate in terms of sarcasm. In his opinion the business in its present aspects is quite, quite nutty.
All of which must be understood to appreciate Dick Rowland’s blistering epigram when informed that “the Big Five” were about to withdraw from all managerial affiliation, and lead the industry in a campaign of “direct to the ultimate consumer” slogans.
“Ah!” observed the president of Metro, dryly, “the lunatics are now running the asylum!”

6-: From Highways and By-Ways of Lil’ Ol’ New York, by Raymond Carroll, published in the Indiana Daily Times (Indianapolis, Indiana, USA) of Monday 29^th May 1922 [page 3, column 5]:

Coming through the door of the eatery was a short, slender, lively young movie magnate, smooth-shaven, with eyeglasses and not much hair on the top of his head—Richard A. Rowland, who said not long ago the motion picture industry “was not a business but a poker game set to music.” He is exceedingly witty. When the screen artists talked of going into the production business for themselves, Mr. Rowland remarked “that would be good—the lunatics taking charge of the asylum.”

7-: From The Sioux City Sunday Journal (Sioux City, Iowa, USA) of Sunday 9^th March 1924 [page 21, column 6]:

Picture Producer Gives “Egotic” Wings

“Egotics” is the latest word to describe motion picture producers who squander money and turn out trash, and the word gathers emphasis when it is recalled that Richard A. Rowland, production manager of First National, also originated the phrase:
“The lunatics are running the asylum.”

8-: From What’s wrong with U.S. talkie stars?, by W. A. M., The Daily Chronicle cinema correspondent, published in The Daily Chronicle (London, England) of Wednesday 7^th August 1929 [page 9, column 4]:

Film Revolt.
The silent film was killed by the suppression of personality.
It was in revolt against the suppression of personality that, early in 1919, Douglas Fairbanks conceived the idea of United Artists.
With Mary Pickford, Charles Chaplin and D. W. Griffith he sought artistic liberty.
At the time a certain movie magnate said of this event: “The lunatics have taken charge of the asylum,” which may now be quoted as showing how utterly out of touch the picture world was with its public.

9-: From Film-Queening At £160 An Hour, by W. A. M., published in the Sunday Dispatch (London, England) of Sunday 12^th April 1931 [page 4, column 3]:

When United Artists was created, a film magnate made the historic remark: “The lunatics have taken charge of the asylum.”
The “lunatics” have done pretty well! Their wing of the “asylum” contrives to flourish, so much so that one of its original founders, Douglas Fairbanks, for his work in “Reaching for the Moon” was paid at the rate of £2,000 a day and collected £60,000 in all. Mary Pickford, another founder-member of United Artists, is said to have got a great deal more for “Kiki.”

10-: From Intelligent Chief Foes of Society, Canadian Claims, by Norton McGiffin, published in The Sunday News and Tribune (Jefferson City, Missouri, USA) of Sunday 1^st April 1934 [page 14-A, column 4]:

A professor of psychology at McGill University in Montreal advances the naive theory that society’s worst ills are caused not by the mental defectives of the world but by the superintelligent. Dr. W. D. Tait claims that “the greatest menace to civilization really is the man who is just a bit smarter than the rest.”
[…]
In drawing up charges against the intelligent of the world, Dr. Tait took in a lot of territory. He insisted among other things that Adolph Hitler and his choice coterie of fellow Nazis were highly intelligent men. A great many people who have never been permitted to collect university degrees will be inclined to differ with the learned professor. They feel that Hitler is a moron, Goering that and worse, Goebels [sic] a goofy broadcaster. They are quite sure in their hearts also that the lunatics are running the asylum across the Rhine when they try to chase out of the country any German whose eyes do not happen to be blue.