From The Miron Blog

For a second, forget the skyscrapers and subway lines and imagine early New York City, an era before the innovation of the grid plan when chickens roamed freely above 14th Street. These buildings are the oldest surviving in Manhattan, all constructed before the 19th century and the beginning of New York's transformation into the city we know today.

Fraunces Tavern (1719)

This historic tavern, restaurant, and museum asserts itself as Manhattan's oldest surviving building. The building, which was originally built to be a home, was sold to the tavern's namesake Samuel Fraunces in 1762, and he turned it into a pub called Queens Head. It was an important hub of Revolutionary activity, and the Sons of Liberty used the space as a meeting place. After the United States was established as an independent nation, Fraunces Tavern housed the Continental Congress until the U.S. Capitol was built.
[Note: Although the Tavern is currently closed for renovation, the Fraunces Tavern Museum at the same location is still open.]
54 Pearl Street; 212.968.1776; frauncestavern.com 

Morris-Jumel Mansion (1765)
Located in Washington Heights, the Morris-Jumel Mansion is the oldest surviving house in Manhattan. During the Revolutionary War, the estate served as a headquarters for both British and American troops. There have been reports of hauntings under the mansion's monumental roof.
65 Jumel Terrace; 212.923.8008; morrisjumel.org

St. Paul's Chapel (1766)
The modest Episcopalian chapel has seen it's fair share of New York City history, from the inauguration services of George Washington to the tragedy of September 11, 2001. St. Paul's Chapel was granted landmark status in 1960, partly because it is the oldest public building in continuous operation in New York City today. Some New York City trivia: St. Paul's Chapel survived the Great New York City Fire of 1776, when Wall Street and its surrounding area burned during the British capture in the Revolutionary War.
209 Broadway; 212.602.0874; saintpaulschapel.org