I've found reference (Oxford Dictionary National Biography) to the interment of noted Quaker minister Daniel Wheeler being at the Friends' Burial Ground, Orchard Street, New York City. I am unable to find any reference to this cemetery. The location, if correct, should not be hard to find in historical records since Orchard Street spans only 7 1/2 blocks on Manhattan's Lower East Side from Division Street to East Houston. Having walked that street many times, I never noticed any plaque mentioning such a cemetery once existed.
A mapping project of historic Manhattan Cemeteries (including those no longer existing) does not show this cemetery. It only shows a Moravian Cemetery (disinterred and moved to Staten Island) that was once entered from Orchard between Delancey and Rivington.
The only possible alternative I see is the Quaker Little Green Street Cemetery (bounded by Liberty and Little Green Street which is now Liberty Place). F-a-G has that here.
Any assistance in finding further information is appreciated.
Edit: if you are not familiar with the story of Daniel Wheeler, the short profile of the Quaker farmer of Tsar Alexander I is worth the two minute read.