Deliberate mistake: The show is set in New York. Joey was a cast member on Days of Our Lives, which is filmed in Los Angeles, but is still always around.
The One With Ross' New Girlfriend - S2-E1
Deliberate mistake: The door to Monica's bedroom is open in many shots, such as when Ross introduces Julie, and we can see that there's a wall (with a framed picture) directly opposite the doorway, but when Phoebe's giving Monica the haircut they're in Monica's bedroom and there's no such wall in front of the door.
The One Where Chandler Doesn't Like Dogs - S7-E8
Deliberate mistake: When Phoebe is trying to sneak the dog out of the apartment, the guys say that they have noticed that the bag is moving, yet the bag doesn't move at all. (00:04:10)
The One Where Chandler Doesn't Like Dogs - S7-E8
Deliberate mistake: Monica sees the dog on her settee from Ross's apartment, which wouldn't be possible from the angle as Ross's place is lower than Monica's. (00:04:50)
The One With Phoebe's Birthday Dinner - S9-E5
Deliberate mistake: Across the whole show, whenever Ross closes his apartment, the door remains unlocked; but in this episode, when he and Rachel go outside, the door locks itself. (00:07:00)
Answer: The painting doesn't have a name, and if it does it is never mentioned in this episode.