The New York State Newspaper Project indicated the Lansingburgh Times published into 1904 but found no holdings in New York after the May 25, 1895 issue. References to it in contemporary publications do suggest that it may have been published at least until 1902.

The Lansingburgh Times was designated as the official paper for the ensuing year.
“Lansingburgh; Village Trustees and Various Matters.” Troy Daily Times. March 25, 1896: 5 col 2.

Lansingburgh Times, established by Alexander Kirkpatrick.
Anderson, George Baker. Landmarks of Rensselaer County, New York. Syracuse, NY: 1897. 396. https://books.google.com/books?id=TuNPAQAAIAAJ&pg=PA396

When the board reconvened Mr. Patchke offered a resolution, which was adopted, designating the papers to print the official canvass as follows: Rensselaer Gazette and Sun, Troy Advocate, Troy Observer, German Free Press, Lansingburgh Times, Rensselaer County Herald.
“The Canvass.” Troy Daily Times. December 23, 1898: 3 col 2.

Lansingburgh Times. advertisement. Troy Directory. 1902.

Lansingburgh Times. advertisement. Troy Directory. 1902.