Perhaps surprisingly, there has been no serious attempt at eliminating Central Park since it opened in 1857. The park was an immediate hit, particularly for the wealthy, who enjoyed riding their carriages around the park's impeccably crafted drives. As the city pushed northward, helped along by extended omnibus service (and later, by the construction of the Elevated trains and then the subways), other New Yorkers had better access to the Park; indeed, several features of the park were meant to appeal directly to the lower and middle classs, such as a zoo (the arsenal, one of two structures predating the park, opened with an animal menagerie in 1860; a permanent zoo built behind it in 1875) and a dairy to serve fresh milk to children (at the time, access to fresh milk was a big problem, as much of it came from diseased cows and the milk would be left out for days and/or mixed with dirty water).

Quickly, Central Park was becoming an important part of New York City, a wondrous testament to the forethought of city planners in the 1850s (the Commissioners' Plan of 1811 did not provide for much green space, and surveyor John Randel, Jr., planned streets through the entire park; one of his bolts is still embedded in rock in the park!). When the land was seized by eminent domain, more than 1,600 people were evicted (including more than 250 residents of Seneca Village, around 85th Street on the west side of the park, a notable community of freed slaves and immigrants; the foundations of the village's Church is still visible). Stretches of Eighth Avenue and 59th Street were renamed, respectively, to Central Park West and Central Park South and soon became some of the most desirable real estate in the city (along with Fifth Avenue, allowed to keep its prestigious name).

Given the city's love affair for Central Park, its rich history, and its vital position as a lush green space in the middle of an incredibly dense island, I don't think any consideration to have it torn down (or even significantly altered at this point) would have been taken seriously. Early investors such as Edward Clark (developer of the upscale Dakota, who also bought several row houses for the middle class and a slew of tenements to boot) observed the city pushing northward, and invested in prime real estate (particularly on the Upper West Side, which until the 1870s had been the site of several sleepy hamlets with colorful names like Harsenville and Striker's Bay). Many significant city cultural institutions and museums are in or near the park, such as the Metropolitan Museum (original building 1880), the Museum of Natural History (original building 1877), the New-York Historical Society (building 1908), the Frick Collection (building 1914), the Museum of the City of New York (building 1932) and El Museo del Barrio (1969) (among many, many others). I could go on at great length about the park's many treasures, including Revolutionary War cannons, the gates from the old Vanderbilt Mansion, fortifications from the early 1800s, the aforementioned Randel bolt and the old remnants of the Central Park reservoir (which brought fresh drinking water into the city as part of the Croton system).

More or less, no, there has never been an attempt at eliminating Central Park. It was actually a largely popular aspect of Manhattan's development.

Manhattan was originally only on the southern-most point of the island, and it slowly expanded northward up the island as the population grew. This lasted until the early 1800's, when the Commissioner's Plan of 1811 was written to create the grid system for a logical and controllable expansion. That is why southern manhattan is a bit of a mish-mosh of streets, while it becomes a standard grid at about 14th street. The Commissioner's Plan, though did not account for Central Park.

At this time, New York was already a bustling metropolis and had an enormous population. Similarly, it was not "New York City" as we know it today, and Brooklyn, Queens, Bronx and Staten Island were not yet incorporated and were independent areas. So the population of New York (aka only Manhattan) was growing in terms of both industry and population, with little to no free space. Transportation off the island was reliant on ferries.

Many New Yorkers were calling for a public park in mid-19th Century and the City added in Central Park for development, and continued to grow the city around the park.

The only time I could see Central Park being a "burden" on New York was in the early 20th Century, when the Central Park Commission was dissolved, leaving the park with no governing authority. During that time, it had no up-keep and was generally "bad"--full of litter, homless populations, crime, etc. But in 1934, Robert Moses became the NYC Parks Commissioner (and one of the most influential urban planners of the 20th Century) and started on a resurgence and expansion of public parkland in New York City. In the early 1960's, Central Park became a National Historic Landmark and could not be altered afterwards.

I just took a free walking tour of Central Park the other day and my tour guide told me that in the 1940s there were people who wanted to build an airport over the park. Can anyone find any sources for this?