The young man has been searching all over New York for the girl he loves, Eloise Vashner. It might be contended that she is the antagonist or that the great, indifferent city of New York is the antagonist. But that is only the background story. If we focus on the part of the young man's search that takes place in "The Furnished Room," then it seems that we have to consider this oily housekeeper Mrs. Purdy to be the antagonist, since she is the only other character in the story, except for Mrs. McCool, the woman with whom she shares a pail of beer at the very end. There is a conflict between the protagonist and the antagonist. The young man is trying to get important information from her. She has the information he wants, but she has her reasons for withholding it from him. She doesn't want it known that someone recently committed suicide in her building, and she specifically doesn't want the young man to know that his loved one committed suicide in that very room, because she wants him to rent the room and pay her the rent money in advance.
Mrs. Purdy has seen a lot of transients come and go. She is very good at what she does, which is renting rooms. Mrs. McCool compliments her on her talent.
“You do be a wonder for rentin' rooms of that kind."
There is an actual contest between the young man and Mrs. Purdy in the story. He smells the lingering odor of mignonette. He is sure Eloise must have lived in this furnished room. He ransacks the room looking for any trace of her.
And then he thought of the housekeeper.
He grills her about the former occupants of the room he has just rented. But she stonewalls him. Who had the room before Sprowls and Mooney?
“Why, there was a single gentleman connected with the draying business. He left owing me a week. Before him was Missis Crowder and her two children, that stayed four months; and back of them was old Mr. Doyle, whose sons paid for him. He kept the room six months. That goes back a year, sir, and further I do not remember.”
She is certainly a smooth liar. He goes back to the awful room and commits suicide himself. Mrs. Purdy has won. She might be said to symbolize the greedy, pitiless city of New York--or any other big city in the world. So a reader would be equally justified in saying that the antagonist in O. Henry's story is New York or Mrs. Purdy.