The One With Joey's Fridge - S6-E19
Revealing mistake: In the closing scene where Ross and Elizabeth are dancing on the beach, you can see that the sky in the background is actually a picture, and you can see where it ends. (00:21:20)
The One With Mac & C.H.E.E.S.E. - S6-E20
Other mistake: When we see the chess board at Central Perk that Ross and Chandler are using, the black pawn in front of the black knight nearest the camera is missing, and from the arrangement of the white pieces, it cannot have been captured yet.
The One With The Joke - S6-E12
Continuity mistake: In the coffee house scene when Joey reveals that he is working there, the strap on Joey's apron goes back and forth from being twisted around his neck and then straightened out. (00:10:50 - 00:12:50)
The One With The Apothecary Table - S6-E11
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. (00:06:05)
The One With Joey's Fridge - S6-E19
Visible crew/equipment: In the scene where Rachel introduces Sebatian to Chandler, Monica and Phoebe, there is a visible boom mic at the top of the screen in the wide shot, right after Chandler says, "Turns out he is kinda funny".
The One With The Apothecary Table - S6-E11
Continuity mistake: When Phoebe and Ross are talking about Ross buying his sheets at the flea market, Phoebe says, "Ross, come on, you gotta loosen the purse strings a little" with her hands by her side. The camera cuts and her hands are now out in front of her.
Continuity mistake: The white dog that Joey gives Chandler sometimes has red lips and sometimes does not.
The One With The Ring - S6-E23
Plot hole: In the opening scene Phoebe and Chandler discuss how Phoebe knows Chandler is going to propose: by walking in on him whilst he was looking at a ring catalogue on the toilet. However, in the previous episode Chandler leaves the apartment right after he found out about Monica booking the wedding slot (on the answerphone) and doesn't come back until he 'freaks out' at Monica and then it is revealed that it was just put on and Phoebe is in on the plan. It doesn't add up, as Chandler would have had to have been looking at the catalogue between hearing the message and deciding to propose, in the same afternoon when he wasn't even in the apartment.
Suggested correction: Chandler decided at some point before the events of TOW Paul's the Man to propose to Monica (and that's when Phoebe walked in). He likely got Phoebe the passes to the museum (because he's the one that said Ross wouldn't be interested). If you look at the episode again, yes, he is freaking out-ish, but this is to fake out the audience. Also, he's "not the groom" because he hadn't proposed yet.
The One Where Ross Hugs Rachel - S6-E2
Other mistake: In this episode, Chandler says that "when it comes to sweets, [Joey] is surprisingly strict" and doesn't allow sweets around the house. Except that in Season Five, "The One With The Kips," we see that Joey leaves Rollo candies everywhere.
The One With The Apothecary Table - S6-E11
Visible crew/equipment: When Rachel and Phoebe are talking in front of the Pottery Barn window, a wooden set wall covered in several signs is visible on the far right in the shots of Rachel.
The One With The Proposal, part 1 - S6-E24
Other mistake: Joey bids on a Catalina 22 yacht, but the picture, and the yacht he and Rachel are on at the end, is a Catalina 18.
The One That Could Have Been, part 2 - S6-E16
Continuity mistake: When Chandler goes into Monica's apartment after he beeps Dr Roger for a "ruptured spleen" his hair is sticking up through the front of his cap. After he tells Monica "You should be with me", the shot goes to Monica and when it goes back to him, his hair is neatly tucked into the cap.
The One With Joey's Porsche - S6-E5
Visible crew/equipment: When the camera follows Phoebe as she is going to clean up the vase she knocked over, the foot of the baby is still visible in the shot and you can see how someone picked it up as the foot goes straight up. Then as the camera moves back revealing there are only 2 babies, you can see a shadow of a crew member moving on the ground. (00:16:00)
The One With The Proposal, part 2 - S6-E25
Continuity mistake: Right after Monica says, "You wanted it to be a surprise", the camera pans to Joey who shuts the door firmly, however in the next shot you can see the door gliding back open and then shutting again. (00:18:40)
The One With The Joke - S6-E12
Continuity mistake: In the scene in the coffee shop where Joey tells his friends about him closing down the coffee shop when he went to the audition, as Rachel says "He left work in the middle of the day to do a personal errand?" two men walk past behind the group. As she continues talking just a few moments later, the same exact men walk past again.
The One With The Apothecary Table - S6-E11
Visible crew/equipment: Right after the "I'll Be There For You" intro, the camera shows the view from outside the "Central Perk" café. As it does so, a couple of stage lights are visible through the upper part of the window, obviously part of the set on the inside of the building.
The One Where Ross Meets Elizabeth's Dad - S6-E21
Continuity mistake: When Chandler is about to put the toilet paper on the table, the picture with the little girl is facing Phoebe. In the next shot, the picture is facing Chandler. Two shots later, it's facing Phoebe again, and then Chandler again in the next shot. (00:08:15)
The One Where Paul's The Man - S6-E22
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).
The One With Mac & C.H.E.E.S.E. - S6-E20
Continuity mistake: As Chandler is writing on the board to remind himself about Joey's audition, the door opens as he's writing "Joey", and there are a bunch of squiggly lines under the y. As Chandler is thinking back at that moment and it moves to present tense, he looks over to the board, and the squiggly lines are different.
The One With Mac & C.H.E.E.S.E. - S6-E20
Visible crew/equipment: A stage light can be seen at the top left hand side of the screen just after Joey slams the door shut after practising a scene with Phoebe. It is visible in front of the blue picture on the wall.