Continuity mistake: When Jack and Judy leave Joey's apartment, there is a birthday banner on the counter that has fallen off. When Monica, Chandler, Joey and Phoebe are racing the toys to see who gets to go, the banner is back on. In this time no-one has fixed the banner as there is not enough time for them to do so off camera and the people on camera don't fix it. It's most clear that it's back on when Monica is saying, "Technically, our dog never crossed the finish line".
Continuity mistake: In the scene when Phoebe says goodbye to David, you can see Phoebe's long hair on her back. The scene goes back and forth between Phoebe and David or Phoebe and Joey (after David leaves). Each time the amount of hair on Phoebe's back changes. Sometimes it is a thin bit of hair, other times a large chunk (while the rest of her hair is framing her face in front).
Visible crew/equipment: After Max tells Phoebe that he and David got the grant and are leaving for Minsk on January 1st, when Phoebe goes to see David, just as she knocks on the door a boom mic can be seen moving at the top of the screen, before it cuts to David's closeup.
Continuity mistake: Just before the opening credits, when Ross says, "That's 'pathet', which is Sanskrit for 'really cool way to live', " Marcel is sitting on Ross's left shoulder in the closeup, but in the next shot the monkey is suddenly sitting on his right shoulder, right before Ross walks off screen.
Other mistake: When Phoebe tells Rachel that she thinks she can help her get over Ross, she starts by saying that she needs some photos of Ross. Rachel immediately starts walking towards the kitchen, walking right past the red box of photos on the table beside Phoebe (which she will pick up on her way back and then hand to Phoebe), before Phoebe even mentions the chocolate (and then glass of water) she also needs. It's obvious that Rachel is headed for the kitchen before she knows she needs to go there.
Continuity mistake: While everyone's at the restaurant Rachel dips a tortilla chip into guacamole and gets some on her hand. When Paolo takes a bite from the chip he also eats the guacamole off Rachel's hand while she's still holding the rest of the tortilla chip, but in the next closeup Rachel's hand is empty, then in the next shot the remaining tortilla chip reappears in Rachel's hand.