
The One With The Flashback - S3-E6
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. (00:19:35)
The One With The Flashback - S3-E6
Continuity mistake: When Ross and Phoebe are in the bar and get on the pool table to start kissing, Ross moves the pool balls out of the way, but two remain. As Phoebe lies down, the 9 ball is next to her left shoulder. The shot cuts closer and it is the 13 ball, then cuts back and it's the 9 ball again. (00:19:40)

The One With The Racecar Bed - S3-E7
Continuity mistake: In the scene where Joey's teaching soap opera for actors, his name changes from the 'a' being on the line to being above it. The line itself also reaches the far right of the board in some shots, and is short of it in others. (00:06:00)
The One With The Racecar Bed - S3-E7
Continuity mistake: When Monica and Rachel show Ben them banging their heads against the wooden beam in the kitchen, Phoebe is hovering on the phone in the background when she is supposed to be at the dentist's office. (00:14:13)

The One With The Giant Poking Device - S3-E8
Continuity mistake: At the start, Phoebe snaps at Rachel that she will go to the dentist. When she says this, her grip on the apple in her hand changes between shots. (00:02:30)

The One With The Giant Poking Device - S3-E8
Continuity mistake: When Ben says Monica's name she's holding the Minute Maid juice box, but in Ben's closeup the juice box is on the table, though it's back in Monica's hand in the next shot. (00:08:50)
The One With The Giant Poking Device - S3-E8
Continuity mistake: At the end when Chandler is singing "Endless Love", Phoebe comes in and when she walks over to Chandler he has his arms wrapped around the Lionel Richie record with the shoe in his hand in front of the record. When the shot switches to just Chandler, his right arm is resting on the arm of the chair and his left arm is holding the shoe behind the record. (00:21:00)
The One With The Giant Poking Device - S3-E8
Continuity mistake: When Chandler is breaking up with Janice and she walks away trying to shake him off, her jumper is untied. When Chandler says "You can't go, I have your shoe" we cut back to Janice and her jumper has been tied at the bottom.
The One With The Giant Poking Device - S3-E8
Continuity mistake: When Rachel and Monica are stressing about the bump on Ben's head, Rachel's hair keeps moving from tucked behind her ear to hanging loosely in her face between shots.

The One With The Football - S3-E9
Continuity mistake: When they are outside for the first time, you see a shot from behind Monica, Joey, and Chandler. Monica then says, "Bring it in," and the shot changes. However, all 3 are now in different poses to what they were in before. (00:04:40)

The One With The Football - S3-E9
Continuity mistake: When Rachel asks Ross if not catching Joey was part of his football strategy, she has her hands outstretched in one shot and then on Ross' shoulder in the next. (00:07:10)

The One With The Football - S3-E9
Continuity mistake: When Joey introduces himself to the Dutch lady, the way and where he holds the football changes between successive shots. (00:08:40)

The One With The Football - S3-E9
Continuity mistake: When Joey tells Chandler to prepare to feel very bad about himself, he has the football in his right hand. When the shot changes, it's in his left. (00:10:30)

The One With The Football - S3-E9
Continuity mistake: Chandler says he's been preparing for something his whole life. When he says this, his right sleeve is rolled up. Camera changes, and now it's rolled down. (00:10:40)

The One With The Football - S3-E9
Continuity mistake: At the end, Chandler points at Joey, and when he does so, his sleeve goes from rolled down to rolled up instantly. (00:22:10)
The One With The Football - S3-E9
Continuity mistake: When they are talking about the Geller Cup, Monica is looking at Ross and then looks down. But when the camera angle changes, she is still looking at Ross and then looks down again.

The One Where Rachel Quits - S3-E10
Continuity mistake: At the start of the scene in which Ross breaks the little girl's leg, we can see that the stairs are just separated by railings. Later in that scene, when Ross and Chandler start walking up the stairs, there are walls on both sides of them. (00:01:40)
Suggested correction: This isn't completely true. In the first shot, we see railing in the middle, as shown in the screenshot. Then, the shot changes to upstairs, looking down, and we can see the wall, as shown in the bottom picture. However, at the end of the wall, it goes into the railings, which is visible just as the pair start to climb the stairs. A screenshot of this can be seen here: https://ibb.co/bmZ8xTc, making it easier to understand.
The problem with the screenshot is the right side is circled to show there's no railing, but the railing is on the left side (to the character's right), but Chandler is blocking the view of the railing. Ssiscool's screenshot is before Chandler starts to walk up the stairs, which is why it's seen.

The One Where Rachel Quits - S3-E10
Continuity mistake: The gang enters Monica's apartment and it is full of dead Christmas trees. After everybody has entered the apartment we see that the apartment door is shut; the shot then changes to the gang around the sofa, but in the next shot when Rachel answers the phone we can see behind her that the door is open. (00:19:00)
The One Where Rachel Quits - S3-E10
Continuity mistake: When Rachel's new boss is explaining about coffee, his head moves from the front to the side with no time to do so. (00:20:30)

The One Where Rachel Quits - S3-E10
Continuity mistake: In the very first shot of Chandler and Rachel sitting on the couch, Chandler has his legs crossed left over right, but it cuts to the closeup and his legs are crossed right over left, then it's back to left over right.