How does the setting increase the conflict between the main characters in Roman Fever?

There could be more than one argument to this answer. I would argue that the place, Rome, is most significant because it is a place that the two mothers (Mrs. Slade and Mrs. Ansley) travelled to together when they were young women. At first, the women discuss how beautiful the place is. Mrs. Ansley seems to have a particular connection to the place; she doesn't explain her reasoning until the very end of the story. Mrs. Slade remarks,

"After all, it's still the most beautiful city in the world.

Mrs. Ansley responds,

"It always will be, to me," assented her friend Mrs. Ansley, with so slight a stress on the "me" that Mrs. Slade, though she noticed it, wondered if it were not merely accidental . . .

At first, they ponder the beauty of the city as they sit in a restaurant during the day. The waiter asks them if they will stay for dinner, and, since both are widows and their daughters are occupied, they decide to stay and observe the city together:

The headwaiter, bowing over her gratuity, assured her that the ladies were most welcome, and would be still more so if they would condescend to remain for dinner. A full moon night, they would remember.

This is an early mention of the moon, which becomes a significant idea later in the story. They continue to discuss the moonlight. Mrs. Slade remarks,

"Moonlight, moonlight! What a part it still plays."

Soon, readers hear that the women had been neighbors in New York City for many years:

Mrs. Slade and Mrs. Ansley had lived opposite each other actually as well as figuratively for years. When the drawing-room curtains in No. 20 East Seventy-third Street were renewed, No. 23, across the way, was always aware of it. . . . Little escaped Mrs. Slade. But she had grown bored with it . . .

Ironically, Mrs. Slade is tired of watching the ways of Mrs. Ansley; she thinks she knows almost everything about Mrs. Ansley's life. However, one of the greatest conflicts of the text comes from the one important secret that Mrs. Slade kept from Mrs. Ansley about her activities on a moonlit night (as a young woman) when they were previously in Rome together.
As the moon comes out while the women sit together at the restaurant, their conversation turns to their memories of that youthful trip. The setting sun is extremely significant in building this conflict:

The long golden light was beginning to pale, and Mrs. Ansley lifted her knitting a little closer to her eyes.

As Mrs. Ansley knits, Mrs. Slade thinks to herself,

"It's all very well to say that our girls have done away with sentiment and moonlight . . . "

Mrs. Slade then instigates deeper conversation about their memories of their youthful trip:

"The sun's set. You're not afraid, my dear?"
"Afraid?"
"Of Roman fever or pneumonia!"

They discuss how Mrs. Ansley got sick from being out at night (in the moonlight) long ago. This leads to a conversation about why she was out. When the reason for her nighttime adventure is revealed at the end of the story, the readers finally understand the depth of the conflict between these two women. Mrs. Slade did not know everything about Mrs. Ansley's life, as she thought she did.