Question: We are attempting to locate a source identifying synagogues and temples no longer functioning or destroyed, in the lower east side of New York City.
Answer: Joyce Mendelsohn's guidebook, The Lower East Side Remembered and Revisited: History and Guide to a Legendary New York Neighborhood (2001) identifies some of the synagogues which are no longer functioning. Oscar Israelowitz's Guide to the Lower East Side mentions both former and "current" (in 1991) synagogues in walking-tour format.
You could compare these brief lists with the extensive lists of synagogues in the American Jewish Yearbooks of 1899/1900 and 1907/1908 [see the "Directory of Local Organizations" section]. The Yearbook volumes list synagogues (and their addresses, rabbis, and officers, activities) in existence at the time of publication. Another extensive list of synagogues, dating from 1917/18, is in the Jewish Communal Register of New York City (1919).
In addition, you may be interested in David Kaufman's "Constructions of Memory: the Synagogues of the Lower East Side" in Remembering the Lower East Side: American Jewish Reflections, edited by Hasia Diner et. al. (2000), p. 113-136.