“Burnham’s Folly” is frequently said to be an early nickname of Manhattan’s Flatiron Building at 175 Fifth Avenue and 23rd Street, completed in 1902. The Flatiron Building was designed by the architecture firm of Daniel Burnham (1846-1912), and, suppposedly, the “Burnham’s Folly” joke reflected the early belief that the building would topple over after a strong wind.
 
However, the nickname doesn’t show up in the New York (NY) Times archive. Google News Archive and the Library of Congress’s Chronicling America also do not show a single citation of “Burnham’s Folly” in connection with the Flatiron Building.
 
A 100-foot tall wooden tower was constructed in Pine Grove (Niantic, Connecticut) in the 1880s by L. H. Burnham. When this structure fell over in the wind in 1914, newspapers described it as “Burnham’s Folly.”
 
“Visitors at first shunned the rooftop observation deck, predicting that strong winds would soon topple ‘Burnham’s Folly’” was cited in a New York City guide book in 1988. New York CIty’s famous “folly” nickname is “Fulton’s Folly,” a moniker given to Robert Fulton‘s 1807 steamboast, Clermont.
 
   
Wikipedia: Daniel Burnham
Daniel Hudson Burnham, FAIA (September 4, 1846 – June 1, 1912) was an American architect and urban designer. He was the Director of Works for the World’s Columbian Exposition in Chicago.
 
Wikipedia: Flatiron Building
The Flatiron Building, originally the Fuller Building, is a triangular 22-story steel-framed landmarked building located at 175 Fifth Avenue in the borough of Manhattan, New York City, and is considered to be a groundbreaking skyscraper. Upon completion in 1902, it was one of the tallest buildings in the city at 20 floors high, and one of only two skyscrapers north of 14th Street – the other being the Metropolitan Life Insurance Company Tower, one block east. The building sits on a triangular block formed by Fifth Avenue, Broadway and East 22nd Street, with 23rd Street grazing the triangle’s northern (uptown) peak. As with numerous other wedge-shaped buildings, the name “Flatiron” derives from its resemblance to a cast-iron clothes iron.
 
The building, which has been called “[o]ne of the world’s most iconic skyscrapers, and a quintessential symbol of New York City”, anchors the south (downtown) end of Madison Square and the north (uptown) end of the Ladies’ Mile Historic District. The neighborhood around it is called the Flatiron District after its signature building, which has become an icon of New York City. The building was designated a New York City landmark in 1966, was added to the National Register of Historic Places in 1979, and designated a National Historic Landmark in 1989.
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The Flatiron Building was designed by Chicago’s Daniel Burnham as a vertical Renaissance palazzo with Beaux-Arts styling. Unlike New York’s early skyscrapers, which took the form of towers arising from a lower, blockier mass, such as the contemporary Singer Building (1902–1908), the Flatiron Building epitomizes the Chicago school conception: like a classical Greek column, its facade – limestone at the bottom changing to glazed terra-cotta from the Atlantic Terra Cotta Company in Tottenville, Staten Island as the floors rise – is divided into a base, shaft and capital.
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When construction on the building began, locals took an immediate interest, placing bets on how far the debris would spread when the wind knocked it down. This presumed susceptibility to damage had also given it the nickname Burnham’s Folly.
 
Chronicling America
3 March 1914, Norwich (CT) Bulletin, pg. 7, col. 3:
PINE GROVE TOWER FELL ON COTTAGE.
Burnham’s Folly Crushed Three Story Building on Sunday.
The 100 foot wooden tower at Pine Grove, Niantic, known as Burnham’s Folly, went down with a twist and a crash during a burstof wind between 4 and 5 o’clock Sunday afternoon. The tower was completely wrecked. It fell ona cottage and almost flattened that to the ground.
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L. H. Burnham built the tower 30 years ago. He intended charging admission and making it a paying investment. For about a year the tower turned in a profit to its owner. The people failed to be attracted by the views to be taken in from the height of the tower and patronage fell off until the tower became known as Burnham’s Folly. It is owned at present by Joseph Burnham, who occupied the lower section for store purposes. Mr. Burnham contemplated taking it down to remove a menace tolife and property and to build cottages out of the material.
   
Google Books
New York, a Guide to the Metropolis:
Walking Tours of Architecture and History
By Gerard R. Wolfe
New York, NY: McGraw-Hill
1988
Pg. 253:
Visitors at first shunned the rooftop observation deck, predicting that strong winds would soon topple “Burnham’s Folly.”
 
Twitter
Wilfred A. Birco
‏@iWilipeleke
The Flatiron Building nicknamed “Burnham’s Folly” #nyc #newyork #manhattan #travel #building… https://instagram.com/p/5TLCLErW7m/
8:38 PM - 18 Jul 2015
Manhattan, NY
 
The Guardian (UK)
New York’s greatest design icons: from Macy’s balloons to the Yankees logo
The Design Museum asked New York architecture critic Julie Iovine to sum up the spirit of her city in 50 design icons. Here are 10 of her favourites
Julie Iovine
Thursday 3 September 2015 07.08 EDT
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Originally named the Fuller Building, the Flatiron was designed by the prominent Chicago firm DH Burnham & Co (Burnham was the architectural mastermind of the 1893 World’s Columbian Exposition) as the headquarters for the Fuller Construction Company. It was not always so popular with critics, who complained that there were too many windows, leaving no walls for bookcases and desks. And its early nickname, Burnham’s Folly, hints at the temporary reluctance of New Yorkers to accept the inexorable transformation of Madison Square from a posh residential neighbourhood into a hub of commerce. Soon enough, however, the whole area became known simply as the Flatiron District.