Plot hole: Joy and Sadness are stuck outside of the control center. They are trying to figure out how to get back, and encounter maintenance workers who are discarding old memories. The maintenance workers show that they have the power to send memories back up to the control center to be played. Why couldn't they send the core memories that Joy had back up the same way? Better yet, why not use that method to send Joy and Sadness back up to the control center? The director of the film is even aware of the plot hole, and said "Yeah, well then we wouldn't have a third act," before explaining how the idea of recalling memories was added in later, "box[ing] [the screenwriters] in a corner a little bit."
Suggested correction: Even if they do send the core memories up to headquarters, they wouldn't be able to get joy and sadness through the tiny gap that the workers send the memories up. Joy has to be in headquarters for Riley to be happy.
This mistake is large enough that the director is aware of it. I think that more than qualifies it to be on this site.
Suggested correction: Joy is a control freak, she wants to return the memories herself. She doesn't even imagine that the workers can do such a thing.
Continuity mistake: Whenever young Riley is pulling the wagon around the house, Fear stops her at a white cord in an outlet. In the next shot, it is a black cord.
Continuity mistake: At the very beginning of the movie when Riley is born, the shot from behind shows the mother wearing the glasses. In the next shot the glasses disappear.
Continuity mistake: Bing Bong hurts his leg and cries candies. One with a grey wrapper vanishes in the next shot. (00:40:00)
Continuity mistake: On Riley's first day of school, she has a blue notebook on top of some other books on her desk. After her story about Minnesota, she sits down, where the notebook is now under another book.
Plot hole: While it makes for a humorous moment, there is no way Riley (and her emotions) wouldn't have known about the big move from Minnesota far earlier than depicted. She would have known long before that she would be moving to a new home, new school, new hockey team, etc. so the idea that she and her emotions are caught completely off guard are very far-fetched.
Continuity mistake: When the friendship island falls, we have a zoomed out shot that shows a clear path of pink ground leading to the three other islands on the edge of the cliff with none of the memory storage units. This path disappears once we leave that shot. It also shows Imagination land as being right behind the island, roughly where Joy is standing, but she's in the memory storage convolutions before and after the landscape shot. (00:36:54)
Other mistake: When Riley and her parents arrive at their new home in San Francisco, several pieces of mail are sticking out of the mail slot in the door. A letter carrier would not deliver mail to a vacant house, let alone not push it all the way through the slot.
Continuity mistake: During the scene near the end when Joy gives the core memories to Sadness, we see the scenes on the screen which are supposed to be Riley's core memories. The first 4 are correct: Friendship, Family, Goofball, and Hockey. The last scene is supposed to be the last core memory: Honesty, where she reveals a hammer that she broke a plate with. Instead, it is the ice skating scene which Joy used earlier in the movie to replace the nightmare. (00:19:05 - 01:21:18)
Continuity mistake: Joy and Sadness had to pass through Imagination Land before reaching the last three islands. But then later after Joy escapes the abyss of forgotten memories not only are both she and Sadness back at the beginning of Imagination Land but that entire area is suddenly sitting right outside Family Island, and then on top of that Joy is able to run around the outside of Imagination Land with a clear and open path to Family Island.
