So far, the worst thing that has occurred in the tour’s history, according to Judy Stanton, a former executive director of the association who has been involved with the tour for three decades, was when a woman tripped down a staircase and dislocated a shoulder a few years ago. Even after the woman was taken away in an ambulance and the house was removed from the itinerary for the rest of the day’s event, the tour marched on, Ms. Stanton said.

But it is the mere specter of exposure on Instagram and in other corners of the Internet, not whether such a thing has actually happened, that has scared people away from volunteering their homes for the tour.

“It’s a strange kind of privacy concern that people have because on the one hand, they are active in social media, in ways that I don’t think are at all private, and then they don’t want people inside,” said Ms. Stanton, who put her house on the first official tour in 1985, when her children were little.

“I am of the generation that rings the doorbell for a cup of sugar,” she said, adding, “I don’t see that kind of culture prevailing in 2016.”

Real estate websites like StreetEasy and Trulia have made it relatively simple to search for a home’s sale price, floor plans and even its owner’s identity, which may have turned off new participants, said John MacIntosh, who runs an investment bank that serves nonprofits and volunteered his house for the tour in the past. Visitors “see the books you read, and they see the bed you sleep on, and I think that people are fine with that,” he said. “But I guess they are less fine about people knowing who they are in a LinkedIn, Internet, social media sort of world.”

For some homeowners, the possibility of widespread social media exposure prompted concerns that their homes would become easy targets for burglars.

Yet organizers of house tours in other parts of New York City say they have not heard the same concerns raised that led to the demise of the Brooklyn Heights tour, offering explanations with a whiff of neighborhood rivalry.