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.
Continuity mistake: Phoebe and Ross are in the soon to be closed bar sitting on the pool table. He clears it of everything, but a few seconds later he stops making out because the "damn balls are in the way" and somehow magics up a handful of snooker balls. If you look closely when he's clearing the table, there are two dark balls at the side which he studiously ignores, and which Lisa Kudrow then surreptitiously grabs and takes onto the table with her so David Schwimmer can grab them later.
Other mistake: In series 8, 'The One with the Videotape,' when they are watching Ross and Rachel on the TV and when Rachel (on the video) is discussing the invitations and how Monica didn't shut up about sending them, Chandler points at the TV and makes a kind-of confused face. A few minutes later, just before he says 'Did you do it on our invitations?' the same shot of him pointing and pulling a face comes up.
Continuity mistake: Right before Ross and Rachel storm in, the girls show Joey how much their waxing stuff hurts. Phoebe starts putting the wax on Joey's inner arm, but then the shot changes and she is spreading in on the outer side. Also she slaps the strip on and the blue tab is hanging down off his arm, but then the shot changes again and it is on his arm with the other end of the strip hanging off the other side.
Revealing mistake: When Monica offers Joey lemonade and he takes off his clothes and is behind the lamp, the tops of his boxer shorts are visible when he says "will you just gonna give me lemonade?" Broadcasts may crop off this bottom part, DVD doesn't. You can also see his gray boxer shorts through the plant in the previous shot.
Visible crew/equipment: In the scene where Ross and Joey are explaining phase two of Chandler's break up (going to strip clubs), the camera switches from the girls to Ross and Joey. Behind them is Monica's kitchen window where you can briefly see a reflection of someone passing by (crew member). At first I thought it could have been Chandler, since he was not in the scene, but after a few moments you see that Chandler has been in the bathroom the whole time.
Continuity mistake: In the opening scene Rachel is holding a white felt pen in her hands and gesturing with it while she talks about the soap opera digest magazine. Between shots the pen sometimes has its cap on and then it's off again so you can see the blue tip.
Visible crew/equipment: During the scene when Phoebe breaks into the other Phoebe's house and gets caught, the older Phoebe was telling the younger one that the man in the photo is not Frank Buffay, but is Chuck Mangioni. Right after that, as Phoebe 2 walks towards the camera, you can see that the door behind her moves open, and there is a visible shadow of someone who walks by that doorway.
Factual error: When Pete takes Monica to Rome for the date this would have been impossible as Rome is 6 hours ahead of New York and at least 4/5 hours travel time, assuming they went by Concorde (double that otherwise). They would have had to have left New York early in the morning to make dinner for 8/9 pm in Rome.
Continuity mistake: When Rachel is talking to Joshua near the drinks table at the party, he is holding a blue cup in his left hand and nothing in his right (this is most obvious when she's offering him a cherry). When she begins to choke, he raises his empty right hand to pat her on the back, then when she walks away there is suddenly a bottle in his raised hand. The shot changes, and his hand is again empty and he reaches down to the table and picks up the same bottle.