Deliberate mistake: When Faramir is about to release Frodo and Sam, not only are the Elven brooches the wrong way round, as someone else has noted, but the whole scene is flipped. You can tell because the leather straps that hold their baggage are the wrong way round as well. Faramir's shots go from normal to flipped a few times, when he walks over, kneels and then rises. (02:40:30)
Revealing mistake: When the women and children are evacuated to the Glittering Caves under/behind Helm's Deep, in the first shot, a hooded person leans on a stalactite making it sway back and forth. (02:42:00)
Continuity mistake: When Theoden is spouting his poem about the horse and the rider, Gamling is putting the armour on his King but when the camera returns to a wide shot, he still only has his breast plate on. (02:42:00 - 00:55:35)
Deliberate mistake: In the scene after the Helm's Deep battle, when Aragorn meets Eowyn, the earlier footage of Aragorn's return to Helm's Deep is used. He has the wound on his shoulder and is dressed like he was during the 'falling' episode. The background (people in the fortress walking calmly) also indicates that this takes place before the battle. This scene should immediately follow the one where Aragorn is greeted by Gimli ("you are the most reckless man I ever knew"). Confirmed on cast commentary, extended DVD. (02:45:20)
Revealing mistake: When the men are retreating into the halls of Helm's Deep, you can see an archer in the bottom right corner repeatedly shooting his bow without an arrow on the string.
Revealing mistake: When the Ents are releasing the river, on the bottom left hand side, you can see an Orc chopping at an Ent. However, when the Ent moves, the Orc is still chopping the same place.
Other mistake: At the very end of the film, where Sam, Frodo and Gollum are approaching Mordor after leaving Osgiliath, it pans up over the mountains to reveal Orodruin (Mt. Doom) in the foreground to the right and The Tower of Barad-Dur in the background to the left. According to Tolkien's maps, as seen in the prologue to the Fellowship of the Ring, if they were approaching Minas Morgul from Osgiliath, Orodruin would be to the left and Barad-Dur to the right.
Other mistake: When Aragorn & Gimli are being pulled up the wall after defending the gate at Helms Deep, Gimli is a puppet (A very good puppet though.) when you see him in Aragorn's arms just before they get pulled over the top by Legolas, only a quick shot but still visible.
Continuity mistake: Helms Deep's gate is destroyed by a battering ram, and shorn up like a barricade while Aragorn and Gimli hold back the orcs. However, when the orcs finally burst through the gate, it has reverted back to a fully intact gate that opens as if on hinges.
Revealing mistake: Just as Theoden yells, "They have broken through, the castle is breached." Legolas is first seen on the exterior wall pulling the string of his bow with no arrow in his hand twice, but the other times he pulls the string on the bow we do see the arrow before reaching the bridge. Also, after he crosses the bridge, before he gets to the stairs, when he reaches behind him for an arrow the arrow is on his back, not in the quiver, it's just simply on his back.
Continuity mistake: Aragorn, Legolas, Gimli, Theoden and others are in the Keep after the walls have been breached, while Aragorn is talking to Theoden about riding out together and the like, Legolas and Gimli are tipping tables up and putting them against the door. However, in one shot when Aragorn is asking about a passage to get the women and children out look behind him, Legolas and Gimli are standing with their weapons drawn, then in the next shot they're back to tipping tables up.
Continuity mistake: As the Uruk-hai charge the walls of Helm's Deep, the elves release their arrows first, no problem here. However, when the Rohirrim shoot, the arrows that strike the enemy are not the short, brown-leather fletched arrows that they should be using, but the long white-fletched elven arrows.
Continuity mistake: When the Uruk-hai are being attacked by The Riders of Rohan you see Pippin on the floor with some mud on his face, yet in the next shot when he is almost crushed by a rearing horse he has got no mud on his face.
Revealing mistake: In the final shot of Gandalf and the Balrog's plunge into the underground lake, a seemingly solid rock in the top-right hand corner disappears for a moment, leaving a black void.
Continuity mistake: When Gollum and Sam and Frodo are by the river and Sam and Frodo are talking about Gollum. Frodo turns around and yells something like, "What do you know about it?" When he yells this, he and Sam's faces are a few inches apart. When it cuts to both of them, they are about a foot apart.
Continuity mistake: When Aragorn, Legolas and Gimli meet Gandalf the White in the Fangorn Forrest, Gandalf says that Hobbits whose tracks they are tracking "Have passed this way the day before yesterday." It cannot be so, because the day starts when Gimli says, "Three days' and nights' pursuit," Legolas says that blood has been spilled last night, and later Eomer confirms that they slaughtered the band of Uruk-hai during that night. Therefore, Hobbits could only have passed that way yesterday, more correctly during the night. Aragorn, Gimli and Legolas meet Gandalf immediately after they enter the forest, so it must be the same day (we know this because the forest is unknown to them; remember Aragorn telling Gimli to lower his axe). However, it fits in Gandalf's speech in Fangorn so beautifully, so it must be deliberate.
Continuity mistake: When Aragorn is lying on the couch in Arwen's home, during the dream/flashback sequence, he at first has the edge of a grey/green pillow visible just above his right shoulder. After a cut to Arwen and then back to him again, the pillow has slipped down so that it cannot be seen behind his shoulder, though he hasn't moved yet. When he stands, the pillow can be seen on the couch.
Continuity mistake: When Theoden says "I will not risk open war", you can see Aragorn in the background taking the pipe out of his mouth, which he does again when he says "Open war is upon you, whether you would risk it or not".
Revealing mistake: The scene at the top of Saruman's tower when he is looking down watching the Ents is shown in reverse. His hair is the giveaway, blowing the wrong way. After cutting away to the Ents once more, they reversed the film again, when he is walking backwards.
Continuity mistake: When Theoden is going to kill Grima on the stone stairway, behind him are Gandalf and three of four guards that take up the whole width of the stair way. Right when Theoden is about to strike, Aragorn pops out of nowhere and stops him. How could he have gotten there in time if we didn't see him push through the other guards, or see him behind them?
Answer: The movie makes it clear that Gandalf has lost some of his memory and his personality has changed. Gandalf actually did die and pass over to the "other side", as it were but was sent back to complete his task. However, he was sent back as a similar, but different entity. Gandalf the White does not have all the memories of Gandalf the Grey, at least at first. He doesn't even remember that he used to be called Gandalf the Grey until someone points this out to him. It is implied that Gandalf doesn't even remember Sam until Aragorn mentions him. Gandalf then searches his memories and remembers who Sam is and his eventual importance to Frodo's quest. When he finally remembers this, he is pleased that Sam went with Frodo, as he will play a crucial role near the end of the journey.