The substitute teacher, Mrs. Dodds, writes, "I understand a fury in your words But not your words" on the blackboard, when the actual line from Othello 4.2 is, "I understand a fury in your words. But not the words".
When Percy is in the Demigod infirmary after the fight with the Minotaur, there is a shield seen leaning against the table next to him as he sleeps. He is then woken up by Grover, the camera angle return to Percy sitting up, and the shield is gone.
In one scene, when Percy is new to the camp, he & Chiron are talking alone by the lake. When the camera is on Chiron, Chiron's arms are folded. But, when the camera switches to Percy, Chiron's arms are suddenly down.
As Percy is entering the museum he turns to see a man watching him from across the street. As the bus goes by, we see glimpses of the man standing and people walking by. When the bus goes by, the man and all the other people are gone. Those entering the scene from the left or right would not have had time to exit before the bus passes.
The heroes escape the Lotus casino by jumping into a Maserati that is offered as a prize and driving off. Due to fire danger the car on display would have had no gas in the tank so it could not have started and run.
Like in many modern-day depictions, the Underworld is depicted as a hellish place full of fire. However, in Greek mythology, all the dead went to the Underworld, their treatment differing on their behavior on earth. So while there was a place where the sinners were tortured, the Tartaros, the Underworld also had the Elysium, which was more or less a paradise for the good souls, the Mourning Fields for people who died while yearning for unrequited love and the Asphodel Meadows for everyone else. Also, the Tartaros was not simply a place of fire, since this is a depiction of hell which only became popular over the last few centuries.
When Percy discovers how long they were inside the Lotus Casino, he says that their deadline is at midnight the next day, after which Zeus and Poseidon will go to war unless the bolt is returned to Olympus. However, the purpose of the heroes' quest was to reach the Underworld to rescue Percy's mother - none of them know anything as to the bolt's location at this point, nor was it their intention to find and return it to Zeus in the first place, meaning the summer solstice deadline shouldn't be an immediate concern to them.
During the Parthenon sequence, after Percy touches down after getting the pearl, the wings on his shoes are absent. This is more notable when we are shown them fighting around the pillars from long shots, shots from behind the men when they are talking together and when he is rolling/running across the ground.
In the opening sequence, Poseidon tells Zeus it has "been many years." In the Percy Jackson series, the Olympians held a council each year on the summer and winter solstices, meaning they would have seen each other in the last six months, but the films make it clear that they do not follow the same continuity as the books.
At the film's climax, Luke claims it was never his intention for Percy and his friends to escape the Underworld alive, yet he gave them the map to the pearls they used to do so. If he hadn't meant for them to succeed, then why did he give it to them in the first place?
During the fight with Medusa, she knocks over a row of statues which fall on Percy, trapping him. As he tries to free himself it is clear that he was not stuck; none of the statues was even touching him.
When Chiron in centaur form walks, he looks strange - but it's not because of bad CGI. It's because a horse, like all four-legged animals, walks with a four-beat gait; whereas whoever animated Chiron clearly didn't know this and tried to force the "walk" into a two-beat, unnatural gait.
When Percy and Annabeth are talking by the pool, Annabeth has a red mark on her arm, which she explains was given by the woman who was turned to stone by Medusa. However, in the earlier scene, there is no trace of the mark.
After Percy beheads Medusa, her body is on the ground as Percy is wiping the sword clean. On her arm is the green pearl in plain sight, but when Percy bends down to pick up her head the pearl is covered by her sleeve.
Both Percy and Annabeth are teenagers and in a real Las Vegas casino would not have been allowed to roam unaccompanied in the gaming area without someone over the age of 21. However, Lotus isn't a normal casino; it is a supernatural entity with its own rules.
When Percy sees the final pearl in the Lotus Casino, on the spinner, it is on slot 33. When Percy picks it up later, it is on slot 15. While the number did change where the pearl was located, Percy was away from the table to rescue Grover and Annabeth. It is highly probable that the dealer collected the pearl and spun again as he would in a "regular casino" to keep up the illusion.
Annabeth is not able to release her arm from the hand grip of the woman turned into stone by Medusa but when Grover cuts off the stoned statue's arm with his sword, Annabeth is able to remove the stoned hand grip by simply shaking it off although it is clear that it was not broken.