Continuity mistake: At the beginning of the episode, Phoebe gives Monica a push on her rollerskates and then we cut to Gunther giving Rachel a mug of cocoa. At this point, Monica is about to crash into Rachel, but as the angle changes we see Phoebe giving Monica a push for a second time.
Visible crew/equipment: In the scene where Rachel is trying to hide Ross from Paul, when he's about to go into Paul's bedroom after he says "Did you really hear a bear?", look at the window behind Rachel when the shot is on her. You can see a large light reflected in the glass and it moves slightly (not just in conjunction with the camera, it moves before the camera shifts).
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". (00:14:05 - 00:17:05)
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. (00:12:15)
Character mistake: When Phoebe brings in her frozen cookie and Monica tastes it to try to determine the ingredients, she says "I definitely taste nutmeg." She then goes on to brag about how that's the difference between a professional and a layman. Except that later when she finds the actual recipe, it's from the Nestle Toll House bag - and that contains no nutmeg.
Factual error: There is no way that a cop and a masseuse could have afforded an apartment in Brooklyn Heights, even in the '90s. (Brooklyn Heights was unaffordable in the '80s.) "Rent control" doesn't explain it either, as a vacant apartment - which is what they are looking for - would be raised to "market rate" if a rent-stabilized tenant moved out. This can't be written off as a character mistake: Gary is not naive or uninformed, and his accent makes clear that he is a New Yorker.