Continuity mistake: When Fili and Kili arrive, the door is left open. We never hear the door close, yet it is closed when Bilbo goes to get the door for Gandalf and the other dwarves. (00:19:00 - 00:19:40)
Suggested correction: So... What's your point? I can, and I'm sure you can, close your door without making a sound. Plus the scene isn't one continuous shot so there is no reason it couldn't have been closed off camera.
Continuity mistake: The mark that Gandalf puts on the door (that looks like a capital F) changes. When Dwalin arrives, there is an additional part of a line on the second line of the capital F. But when Gandalf made it that additional part was not there.
Continuity mistake: The scene where Bilbo first meets Gandalf. As Bilbo says "Not Gandalf the wandering wizard who made such excellent fireworks?", the grip of Gandalf's staff changes between two shots. (00:14:10)
Continuity mistake: In the opening scene with older Bilbo and Frodo, the mailbox is next to the gate. In Fellowship of the Ring, the mailbox was behind the edge of the gate. (00:09:20)
Continuity mistake: At Bag End, Bilbo recovering from fainting, sits in an armchair. Gandalf is speaking with him. Behind Gandalf is an alcove with an unlit candle, which is lit in the next shot, although not illuminating any of the alcove, then it's unlit again.
Continuity mistake: In The Lord of the Rings, when the map of Lonely Mountain is shown, it says "Here of old was Thrain king under the mountain" (referring to an ancestor of Thror and not Thorin's father who has the same name and was never king under the mountain). But throughout The Hobbit, the map instead says "Here of old was Thror king under the mountain."
Suggested correction: Thrain was king after Thorin. The map would have been updated after Thrain's death, and he was subsequently buried there.
Thrain II was king after Thorin I and father of Thorin II (Oakenshield), but as said he was never king under the mountain but was king during their exile, Thror was the one who went back to Erebor to sit on the throne again until Smaug sacked it. The Thrain mentioned on the map is Thrain I (father of Thorin I), who is the founder of the kingdom under the mountain after Durin's folk were exiled from Moria by the Balrog (Durin's Bane, Durin being Thrain I's grandfather).
Also, even disregarding that, it is impossible for the map to have been updated to say a different dwarf's name, because the map is given to Bilbo at the end of the trilogy, and the ending shows a close-up of the map with "Thror" still written on it, whereas in the LOTR trilogy, where we see the exact same scene of Gandalf arriving at Old Bilbo's house (albeit from Gandalf's point of view rather than Old Bilbo's), Gandalf sees the map and it says "Thrain" on it.
Continuity mistake: After Bilbo has written the prologue to his book, Frodo is talking to Bilbo and then heading off to wait for Gandalf's arrival. This contradicts the extended edition of The Fellowship of the Ring, where Frodo is shown to have already left the house to wait for Gandalf before Bilbo begins writing his book (which he begins with a section called "Concerning Hobbits," which he states is "where to begin," indicating that Frodo left the house before Bilbo wrote any of the book).
Continuity mistake: When the Arkenstone is found, the miner who found it has one hand on the wall and his other hand by his side. In the next shot, one of his hands is suddenly reaching for the stone. (00:03:35)
Continuity mistake: Frodo puts a fresh smooth sign on the Bag End gate. In Fellowship, when Gandalf arrived at Bag End, the sign was all tattered around the edges. (00:11:25)
Continuity mistake: The shot where Frodo walks back into Bag End with the mail and the shot where Gandalf knocks on the Bag End door in Fellowship of the Ring are apparently the same setting, but it is different between the two different movies. First, the grass around the bench closest to the door has long grass around it, but there is short grass in Fellowship. Second, behind the bench is some sort of bucket on its own. In Fellowship, there are a few things next to it that aren't even there in The Hobbit. Third, further behind the bench is a lamp post, about a meter and a half away from some pink flowers, that was right next to the pink flowers in Fellowship. Fourth, next to the window are some green flowers that cover half of the windows. In Fellowship, those green flowers are a much lighter green and cover the entire window. Considering only an hour passed between these two times it's hardly possible the flowers would grow, and since Bilbo was too busy working on his story, he wouldn't have moved something. (00:09:30)
