Factual error: When the gang all see the beach house for the first time, it's covered in a layer of sand, which is supposedly left over from a recent flood. Well, if there really had been a flood, there would have been mold on the walls and debris or junk from the outside, and the sand wouldn't have been nice and clean and dry; it would have been moldy. And there would've been a pattern, likely on the walls, of where the water rose up to and where it receded. Not only that, but how did it get in? All of the doors and windows are intact.
Other mistake: In the episode with Ross's wedding to Emily, Joey is talking on a cell phone with Phoebe as he walks down the aisle. Joey holds the phone out to Ross saying 'it's Phoebe', and one can clearly see that the phone's screen is completely blank. The phone isn't even turned on.
Continuity mistake: When Phoebe and Rachel are first talking about the apothecary table, there are two piles of CDs on it. In the side shot you can see that the two CDs on the top of the pile in front of Rachel are overhanging by a few centimetres. A few shots later, however, the top two CDs are overhanging on a different side.
Factual error: When Ross shows Chandler the book, he claims it is in "the biggest library on campus" at NYU, which would be Bobst library on Washington Square Park South. Later, in Central Perk, when Ross tells Rachel and Joey about the people "making babies" in front of his book, Joey says, "Wait a minute. Fifth floor, against that back wall?" Bobst's fifth floor is conference rooms only; there are no stacks on that floor. Joey might have been confused, but Ross should know better, being a professor and having just visited that location.
Continuity mistake: In the coffee house when Rachel is talking to Ross about trust, a couple comes in behind her and stands at the bar. In the next shot of Rachel they have switched positions; then they are gone, and you never see them walk past Chandler.