Revealing mistake: In the scene when they confront the stripper at Chandler's office. Ross starts shouting about his grandmother's ring, and as Chandler says "way to be cool, man", you can see Joey ducking behind Ross, covering his mouth with laughter. A brief cut away, then back and he's standing with the others again looking serious.
Revealing mistake: When Monica and Phoebe are in Central Perk, Phoebe is looking through the list of ideas that Monica has come up with for the use of Phoebe's van. If you look closely (most obvious just after Phoebe says "let's plan the wedding reception") you can see that every single page of the notebook Phoebe is holding has the same thing written on it, instead of lots of different ideas that Monica is meant to have had.
Revealing mistake: When Phoebe is looking at the front page of the "Chelsea Reporter," Monica says her restaurant review is on the back page. However, Phoebe doesn't turn to the back page, but merely flips the paper over, looking at the bottom of the front page (and finding the review there).
Revealing mistake: During the end bit, before we get to see Phoebe when she is being interviewed at Lara and Jeni's Massage, if you look closely at the name of the massage parlor, you'll see that, although the camera shakes and bumps, the words stay on the very same spot on the screen. Obviously the canopy was blank and the name was digitally added later.