Although Johnny Depp lives in France, he doesn't appear in any of the scenes shot in France. All of his scenes were filmed on set or on location in England. All of Judi Dench's scenes were filmed in England. The DVD commentary notes that her husband was very ill during the shoot, and it was important for her to be close to home.
It is Johnny Depp and Juliette Binoche jumping in the water themselves (despite having stunt people). It was reportedly freezing and muddy. But in the close up shot of Juliette's face in the water, the man pulling her is not Johnny. This was shot in a water tank.
Anouk's imaginary kangaroo's name is spelled Pantoufle, not Pantouf. This is clear in the closed-captioning. Pantoufle means "slipper" in French (this makes slightly more sense in the book, where Pantoufle is a rabbit).
The boat on which Vianne and Roux have their love scene was built on a stage and then physically rocked to mimic the water. When the outside of the boat is shown after they kiss, the shadowy figures were added digitally.
The French town of Flavigny-sur-Ozerain, where most of the movie was filmed (except for scenes on the water), does not have a chocolatier. However, it is the home of l'Abbaye de Flavigny; the abbey makes anise (and other flavored) candies that are sold worldwide in little tins decorated with flowered designs and copies of old engravings. Unfortunately, the factory does not allow visitors.
In the scene where Pere Henri is dancing, Alfred Molina was originally supposed to walk down the steps and see him. However, because he actually had a hurt toe, he just stood there instead of walking down the steps.
The film's Oscar campaign became the grounds for a major fight between then-Miramax Chairman Harvey Weinstein and reporter Lisa Schwarzbaum, when Schwarzbaum enraged Weinstein said that the studio had cheaply used the endorsement of Anti-Defamation League Chairman Abe Foxman (who praised the film for its promotion of tolerance) as part of what she called a "P.T. Barnum" campaign. The dispute petered out when each side made vague apologies and the film did get a number of Oscar nominations.
The trivia items below may give away important plot points.
The scene where Alfred Molina (Comte Paul de Reynaud) eats the window display was shot on his last day of shooting in case he didn't feel well after eating all the chocolate. He didn't. Along the way he got a piece stuck in his throat briefly, and we see him lean over and expel it without breaking character.
As the camera is zooming out on the village at the end during the credits, you can see a man walk through the lower half of the square. The director wanted to be able to prove that it was the actual village and not a miniature.
The storyteller, voice of Tatyana Yassukovich, is revealed by one of the last scenes to be the adult Anouk. It was debated about having a well known celebrity take this role, however in the end it was decided that this would take away from the power of the film.