Her building is part of a parcel between 72nd and 59th Streets along the Hudson River that Mr. Trump developed with a group of Hong Kong billionaires starting in the 1990s. Seven of the buildings in the area were emblazoned with his name. But while some of the condominium towers in the development are still managed by the Trump Organization, the three rental buildings now belong to a company called Equity Residential, which also manages them.

Martin McKenna, Equity Residential’s vice president in charge of investor and public relations in Chicago, said the company had “a contractual obligation on the use of the name.” He declined to say how much longer the contract had to run, but added that, “we’re going to assess this when the obligation expires, taking into account concerns by the residents in their petition.”

Erin Kelly, also a resident of 140 Riverside Boulevard, and a signatory to the petition, said that as her son and nanny were leaving the building on Sept. 26, the day of the first presidential debate, they noticed workmen measuring the rain mats and asked what was going on. A staff member told them new, Trumpless mats were being ordered.

“I haven’t had a single interaction with a neighbor or an employee who wasn’t interested in having the name removed,” Ms. Kelly said. “It used to be that we were embarrassed because he was tacky. Now he’s shown himself to be despicable on every level.”

Ms. Jacobs, of 180 Riverside, said one of the doormen and a concierge had told her uniforms without the Trump name were in the offing.

Mr. McKenna, of Equity Residential, said, “We don’t have a comment on that.”

Mr. Trump’s press secretary, Hope Hicks, said that the candidate had not heard of the petition, but that it seemed like “an inappropriate thing to do.”