The Farmer's Market sequence was shot guerrilla style, with permits, but without extras. Most of the people are real shoppers, and many of them didn't notice that Frances McDormand and Jennifer Aniston were among them. However, members of the paparazzi still managed to slip in, trying to get shots of Aniston.
In order to play Aaron #1, Simon McBurney had to keep flying back and forth between Los Angeles, where the film was being shot, and London, where he was directing and acting in a play at the Royal Theatre. He made the trip about four or five times, staying in the U.S. for a couple days to shoot his scenes.
Outside the restaurant where Mike and Olivia have their date, if you look closely you can see a black fuzzy mass on the patio of the House of Pies restaurant across the street. According to Nicole Holofcener on the DVD commentary, those were all members of the paparazzi.
The dinner at the beginning of the film was shot in a real restaurant in Los Angeles. The film crew damaged the interior, leaving marks all over the walls. That scene was the first in which all the key characters were filming together at the same time.
The trivia item below may give away important plot points.
To fulfill obligations with Frances McDormand's contract that she can't be away from her family beyond a certain time frame, the last week of filming was done in New York. These scenes include most of her scenes at home, plus the scene where Jane hits her head on a glass door.