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."manthabeat
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: 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: 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.