When we first see the kitchen, the frying pan handles are sticking out. One of the first things a cook learns is to keep handles at an angle and away so that the pans will not be accidentally hit and cause injury.
Most of the "spices" referred to in the film (rosemary, oregano, thyme, basil) are actually herbs. Spices come from the seeds, roots, stems or fruit of plants, whereas herbs come from the leaves or flowers.
When Remy reunites with his brother outside the restaurant, he offers to steal some food from inside the business. However, Remy already was given a supply of food just a few minutes ago such as grapes and a large chunk of cheese to eat which he had barely started on himself. Why didn't Remy just have Emile help himself to that?
During the press conference, an irritated Remy, hiding in Linguini's toque, yanks on Linguini's hair with his left paw, causing Linguini to hit himself in the left side of the head with one of the two copper pans he's holding up for photographers. However, Linguini has his arms crossed in front of his face at that moment, and thus hits himself with the pan being held in his right arm, not his left. From what we know of how Remy controls Linguini's arms by hair tugging, Remy would have to have used his right paw for that movement, not his left.
When Remy is released from the trunk of the ex-head chef's car, it is clearly daytime. Remy goes across the road to the restaurant to cook for the critic, however, when he arrives, the courtyard is clearly seen to be dark (indicating nighttime), how long did it take him to cross the road?
When Remy is fixing the soup that Linguini ruined, he puts a big wooden spoon in the pot to stir and then to smell. He then places the spoon back in the soup before adding the last ingredient and getting caught. When he jumps from the pot, the wooden spoon is still there, but when Lalo comes over to take some of the soup, the wooden spoon isn't in the pot any more, and doesn't reappear in any subsequent shots.
The poster with the health inspector's picture is even and right next to the light sockets when Alfredo tries to give a pep talk to the crew. When Remy later recognizes the health inspector from the poster, the poster is half-way below the light sockets and lopsided.
In the beginning of the movie, when Remy and Emile (Remy's brother) decide to cook the mushroom on the chimney they are struck by lightening. After flying off the roof and landing on the ground their fur is obviously standing up in all directions with a singed look too it. Seconds later they run into the house to retrieve the saffron for their concoction, but their fur is magically all back in place and smooth as if nothing happened. Nothing occurs in between that would explain how their fur was fixed.
The food critic Anton Ego makes a remark about Chef Gusteau comparing him to Chef Boyardee as an insult. However the real Chef Boyardee(real name: Ettore Boiardi) was a world famous chef for over a decade before his line of pre-made pastas and sauces became popular. He was known for being head chef at the Plaza Hotel in New York City, and even catering president Woodrow Wilson's wedding in 1915.
Gusteau's restaurant was said to have attained a five-star rating previously, yet the French guides don't go any higher then three stars. This detail was most likely deliberately ignored by the producers because many international audiences are used to hotel ratings that go up to five stars, and would not have understood that a three-star rating is the ultimate culinary honor in France.
In the scene where Remy is correcting the soup Linguini had concocted, a 360-degree shot is used to show Remy tossing ingredients into the soup pot. Next to the stove there should be a mop that Remy had used to climb up to escape out the window previously, but is absent during the rotating shot. Instead there is a shadow of the absent mop handle that can be seen.
The goof items below may give away important plot points.
Remy demonstrates that he knows what would happen if he is found out when he maneuvers Linguini into kissing Colette, yet he is incensed when Linguini refuses to acknowledge him during the press conference.
When the restaurant is closed down, the poster on the door says that the restaurant has been closed in violation of French Health Laws. Such cases are handled by each municipality (in this case, Paris), and the poster would state as such. And as the restaurant is in France, the poster is more than a bit redundant.