Continuity mistake: In the scene where Anakin arrives on Mustafar and tells R2-D2 to stay with the ship, he pulls the hood of his robe over his head with two organic hands. His right hand should be mechanical. [This mistake is mentioned in the audio commentary: this shot is taken from a shot of Obi-Wan later in the film when leaving Padme's apartment. Still a mistake, though.]
Continuity mistake: During the Anakin/Obi-Wan fight scene Obi Wan switches lightsabers with Anakin. When Anakin is strangling him we can definitely see that Obi-Wan is holding Anakin's lightsaber in his right hand (note the hilt design), but when the shot is reversed Obi-Wan is holding his own lightsaber in that hand.
Continuity mistake: When Anakin is in the medical centre lying on his back being transformed into Darth Vader, you can see the main part of the mask being lowered into place but Anakin is not wearing the neck section of the mask that it connects to. We then see the main part lowering directly downwards, in the next shot the neck section is in place.
Continuity mistake: In the scene where Anakin returns to Padme's apartment after the fight at the Jedi temple, watch R2-D2 in his compartment in Anakin's ship as the camera angle changes. In some shots, R2's "legs" are flush with the hull of the ship, and in others, they are well above it.
Continuity mistake: When Palpatine is talking to Anakin in his office, there is a table at the end of the hall with a large object in the middle, and a gold cup on each side. When they leave the office and are talking in the hall, and a thin purple lamp/object appears on the end of it for a few shots, and vanishes whenever seen from then on.
Continuity mistake: When the Emperor says "I have waited a long time for this moment my little green friend." Yoda can be seen laying beside his cane on the purple carpet. When they show him standing up, he and the cane are now further back toward the wall on the tan part of the carpet.
Continuity mistake: When Mace Windu and Palpatine fight, Windu backs Palpatine into the corner of the window. Note how close Windu is to Palpatine the whole time. Then from Anakin's perspective, Windu takes a few steps toward Palpatine, his saber pointed at Palpatine. When we saw it from the earlier perspective Windu never took those three steps toward a stationary Palpatine.
Continuity mistake: After Anakin has told Windu that Palpatine is the Sith Lord, Windu gets on his ship which takes off. We see the ship's shadow moving on the ground right next to Anakin. When it cuts the shadow from the ship is gone. When it cuts again to a wider shot we see the ship's shadow suddenly appear again.
Continuity mistake: When Mace Windu and Palpatine are in his office after the lightsaber battle, Palpatine scrabbles into the corner as Windu paces towards him, ending up directly over him with lightsaber pointed at his throat. As Anakin walks in, Palpatine is already backed into the corner and Windu needs a few paces to get to him.
Continuity mistake: In the scene where Plo Koon gets blasted by his Clone Troopers, the exterior shots show he is in an Episode II starfighter (closed/pointed bow, no small rear window on cockpit canopy- best seen as the Emperor gives the order), yet the interior shots of his cockpit show he is inside an Episode III starfighter (forked bow, small rear window on canopy- same as Obi-Wan's starfighter).