This episode is told in flashbacks. Monica's birthday is incorrectly seen as a mistake in continuity. She is 26 in season one. Her birthday party likely takes place during season 5, though it is not shown. During her party, she is already with Chandler. Other indicators, such as Rachael's long hair, indicate her birthday would have happened around season five. So, the continuity of her birthday, and the others' birthdays, is correct.
When Phoebe is at Ursula's apartment, Ursula presents her with a birth certificate. When Phoebe takes it, she is holding it in her left hand. But when the camera shifts to see Phoebe's face, the paper is in her right hand. Then when it switches back to face Ursula, the birth certificate is in Phoebe's left hand.
Phoebe claims to not have (nor even seen) her birth certificate. But we know she has a driver's license which would require her to have her birth certificate to get one.
Joey says: "But it hurts my Joey's apple." and Chandler replies that it isn't named after each individual. But in _"Friends" (1994) {The One Where Heckles Dies (#2.03)}_, Joey emphasizes: "When I first moved to the city, I went out a couple of times with this girl, really hot, great kisser, but she had the biggest Adam's apple." which implies that he does know that 'Adam's apple' is a global term.
In "The One Where They All Turn 30", Rachel mentions that she's "still 29 in Guam." Actually, Guam is 14 or 15 hours (depending on Daylight Savings Time) ahead of New York (Eastern) time. She would have turned 30 in Guam before she turned 30 in New York.
Rachel is celebrating her thirtieth, yet at the end of The One with the Fake Party (1998) she says to Ross: "And you are sitting here with a twenty-eight year old cheerleader". How can she turn thirty, three years later?
At the end of the episode, Chandler says "Well, here we are, just a bunch of 30-year-olds". This is impossible since Joey's birthday is shown as happening before Chandler's. Ross and Chandler are at least a year older than Monica and Rachel, and also, earlier in the episode we find out that Phoebe is actually 31.