Rachel's birthday: In episode 4x05, she says she was born on May the 5th (Taurus) but later, in 7x22, when a policeman asks for her driver's license, he looks at her birth date and remarks that she is an Aquarius (January/February).
Ross' birthday changes several times during the show. In the 1x04, he mentions his birthday being 7 months before October 20th, which would be sometime in March (Pisces/Aries. Soon after he mentions his birthday being in December (Sagittarius/Capricorn), then discloses his birthday is precisely October 18th (Libra) in the later episodes.
On several episodes we are told that Gunther is Dutch. He even speaks some Dutch lines. However, Gunther is a typical German name, and in the Netherlands only people immigrated from German-speaking countries or with one or two German parents are named Gunther.
Rachel's last name is spelled "Green", as seen in Friends: The One with the Cake, on the cake box is written "Green", such as it appears in Warner Bros. website and in the series credits, but in her office door at Ralph Lauren it is erroneously spelled "Greene", as well as on her invitation to Ross and Emily's wedding.
In one episode Monica says how her brother Ross never told her anything about his sex life or when he lost her virginity, but in the episode before, he says how he told everyone, including his sister, when he lost his virginity for the first time to Carol.
In early seasons there are wood boards in Monica's apartment
that go from the floor to the ceiling. In later seasons there are only wood boards by the fridge. In fact, certain directors for the show enjoyed putting their own mark on the episodes they directed. In the case of the visible wooden beam furthest "downstage" it was known as the mark of James Burrows, and would appear every time he directed an episode.
Phoebe's grandmother's taxi, which is meant to be a single vehicle, is portrayed by several cars throughout the show's run, as evidenced by different livery patterns, taxi rates, identification numbers, and features found only on certain years of the Checker A11, such as opposing windshield wipers in Las Vegas (used until 1976) and parallel wipers in every other appearance (1977-82).
Joey and Chandler's apartment has a heat radiator in the living room by the window. In later episodes it is not there. There's also a radiator in the hall just outside their apartment that disappears in later episodes. The wall of the kitchen at the door (at side of refrigerator) there are kitchen gadgets/dishes and food on shelving. That disappears forever in later episodes. The tub in their apartment appears and disappears depending on what is going on. Open bathroom door shows no bathtub at all against the back wall, then shows a sink, then tub with shower curtain for. Chandler playing with the duck in the tub shows it back against the wall again. There's nothing on the wall behind their kitchen sink, then there are shelves, then there is a microwave, no shelves. Monica's apartment has some type of 'awning' over the kitchen area that disappears in later episodes (remember she bonked Ben's head on it and Ross ran into it). Joey claims someone across the way from Monica's apartment with a telescope is looking at him in his apartment. There's no way that could happen based on the view outside that window, plus no way could anyone watch him cooking via a window. Monica never locks her apartment door, yet the day Joey tries to hide from his crazy stalker, Brooke Shields, the door is locked and he and Chandler cannot get in.
Even the show's designer has commented on the fact that the Central Perk sign on the window can be read correctly from both inside and outside the coffee shop. But viewers have noticed that the logo is symmetrical and suggest that it isn't unfeasible for the owners to have placed a sign on the inside of the window which completely covers the one on the outside, making it readable from both sides.
Joey is a regular on "Days of Our Lives" while living in New York. But "Days of Our Lives" is taped in Los Angeles. However, since most characters featured on the fictitious "DOOL" were not actually on the real show, the fact that it's filmed in New York could just be another characteristic to this fictitious version.
In an early episode, it is revealed that the first (and at the time, only) person Ross ever slept with was his first wife Carol. However, several years into the show, there is an episode in which Chandler says that Ross slept with the woman who cleaned their dorm building while they were in college.
In the final episodes when Rachel gets ready to leave the country for Paris for her new job she says how she worked for Ralph Lauren for 10 years, but she didn't start that job until 1999 and the show ended in 2004, therefore she couldn't have worked there for 10 years.
The goof item below may give away important plot points.
Early in season 7, Monica mentions her wedding being on May 15. Ross and Rachel reveal that they conceived their baby about a month before the wedding (April). But as season 8 continues, you see Halloween, Thanksgiving, and Christmas pass with Rachel only having a bump. Halfway through the season, Phoebe mentions that Rachel is four months pregnant. She should be in her third trimester at that point.