The weekend of May 19-20 is the Open House for the Sacred Sites Program run by the New York Landmarks Conservancy. The Conservancy is one of the only organizations that provides funding and technical assistance to religious properties, which often are not eligible for other preservation funding. Since 1986, they have funded more than 660 congregations to help them mobilize their preservation projects.
Because of that, there is probably a church, synagogue or other religious institution near you that has been helped by the Conservancy and is open that weekend for visits and tours. Check this website (see Participant List link under the photo) to see if there is a site in your neighborhood.
Of particular interest is the Church of the Intersession on Broadway at 155th Street. Rarely open except for services, this church was designed by Bertram Goodhue (one of the great collections in Avery Library Drawings & Archives) and is the place where he is buried in a tomb designed by Lee Lawrie, the sculptor who did many works at Rockefeller Center.