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.
Other mistake: To escape from the elevator, Anakin cuts a hole in the ceiling. Later, when they rescue the Chancellor, they almost get crushed by another elevator (it is another one. Each elevator has a number. Obi Wan said the number of this elevator before and it was different than the one that had the hole). However, when a shot of this elevator is shown, it has the same hole that the other elevator has (better seen in slow motion).
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.
Other mistake: During the opening space battle, Obi-Wan's ship falls under attack by buzz-droids, who proceed to cut into his fighter, disabling a number of systems and leading Anakin to try some decidedly dangerous moves to get rid of them. Both Anakin and Obi-Wan are acknowledged experts in the use of the Force and both have been shown to use the Force routinely, even for relatively mundane activities. It's also been firmly established that the Force works on droids and that the Force-user can still affect others through windows, viewscreens and so forth. Yet neither thinks to use the Force to deal with the buzz-droids.
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.
Other mistake: During the scene of Anakin's transformation into Darth Vader, there is a close up shot of his face just before his helmet is put in place. All of his face is shown as being badly burned with the exception of his eyelashes, which are still impossibly there. Subjecting one's face to any heat source hot enough to sear one's face will also burn away one's eyelashes as well.