On that spot, they established the St. John's Burying Ground. Here's a shot from the NY Public Library's archives showing the cemetery while it was fully established. (Note the large monument on the right side of the frame; it's where we'll be going.)
As far as I can determine, the church took the attitude that the city had unfairly appropriated their land and now all of the bodies were their problem. Roughly 250 families paid to have their loved ones moved uptown and the rest were, apparently plowed under.
The new park was designed by the firm of Carrere and Hastings, later famous for designing the main branch of the New York Public Library that stands at 42nd Street and Fifth Avenue. The park they designed, St. James Park was a Victorian garden with lush landscaping, a lagoon, a gazebo and a sunken garden. The only evidence of it having been a cemetery was that they retained (albeit in a different location), the monument I mentioned from the above photo.
For some reason, after only seven years, it was decided that the neighborhood needed a playground instead of a Victorian garden and the park was changed again. The next time anyone was reminded that the playground had once been a cemetery was in 1939 during another round of renovations. Workmen hit an underground vault and unearthed a small casket. The casket had been designed to mimic an Egyptian sarcophagus and had a silver plaque identifying its occupant as Mary Tisdall, aged 6 years and 8 months. She had died on April 14, 1850. According to The New York World-Telegram "The girl's cast iron casket...had a glass window in the top. Her white silk dress still looked fresh and dainty. After 89 years you could still see that she'd been a pretty yellow haired child."
The park has been renovated a number of times since then and renamed for former mayor (and St. Luke's Place resident), James J. Walker. Here's a bit of what it looks like now, viewed from Hudson Street. (Yeah, I just did a screengrab from GoogleStreet.)
Here's the monument as it looks today.
Here are interred the bodies of Eugene Underhill aged 20 years 7 months and 9 days and Frederick A. Ward aged 22 years 1 month and 16 days who lost their lives by the falling of a building while engaged in the discharge of their duty as Firemen on the first day of July MDCCCXXXIV (1834)
The south side reads:
This monument is erected by the members of Eagle Fire Engine Company No 13 in connection with the friends of the deceased to commemorate the sad event connected with their death and the loss which they deplore
I hope Eugene and Frederick and all of their neighbors are sports fans.