Originally, the scene in which the couple is discussing their problems in bed with Dr. Sobel (Billy Crystal) was supposed to include only the woman and Dr. Sobel, but Crystal thought it would be funnier if the two of them were present. The man playing the husband is the head of the sound department for the movie, Les Lazarowitz, who being a heavyset guy made the scene even more comical. He deferred the sound mixing duties to his assistant while this scene was being filmed. Coincidentally, Lazarowitz was the boom operator (uncredited) in the fruit stand scene of The Godfather (1972), which is parodied in Analyze This (1999).
In the movie, Dr. Sobol dreams that he is shot by some gangsters while buying fruits from a fruit stand. This is a parody of a similar scene in The Godfather when Don Vito Corleone (Marlon Brando) was also shot buying fruits from a fruit stand. Coincidentally, Robert DeNiro played a younger Don Vito Corleone in The Godfather 2.
In the original script, there were more scenes in which Dr. Sobel (Billy Crystal) receives gifts from Paul Vitti (Robert De Niro). In one of them, former Yankee great Phil Rizzuto, at the request of his friend Paul Vitti, would give Dr. Sobel his spike shoes because Dr. Sobel is a Yankess' fan, which Billy Crystal is in real life).
The trivia items below may give away important plot points.
The prison scenes at the end were actually filmed at the Sing Sing prison. The prisoners seen walking down the corridor at the beginning of the scene are real prisoners who belonged to the prison's drama club. The shooting at Sing Sing took only one day.
The scene towards the end in which all the Mafia bosses meet was filmed twice. The first filming occurred in Queens, New York. After watching the scene, Robert De Niro offered some suggestions, including that Dr. Sobel (Billy Crystal) "really belt" Jelly (Joe Viterelli). So the scene was filmed again, this time in Miami, Florida. In the final cut, the two scenes were merged. The shots of the scene in which a room full of people can be seen were taken in Queens whereas the shots in which Dr. Sobel is talking to Primo Sidone (Chazz Palminteri) and slapping Jelly were shot in Miami. There are only two or three people in the room for the scene that was shot in Miami.