Anakin decapitating Count Dooku by Palpatine's order.
Anakin using the Force Choke on Padmé. Seeing Padmé pregnant and vulnerable, heartbroken and pleading with him one moment, then choking, then unconscious is horrifying enough on its own, but then Anakin doesn't come down from his evil high afterward. In fact, he gets worse.
Even worse for being deeply unsettling Truth in Television: one of the top causes of death for pregnant women is murder, often at the hands of an intimate partner.
As far as the dark parts of John Williams' score go, almost nothing is as disturbing and unsettling as the track "Palpatine's Teachings". It is one of those kinds of music that can make anything seem evil and frightening when placed over it.
A bit of elaboration for this. This song plays during the Opera scene, where Palpatine tells Anakin about "The tragedy of Darth Plagueis the wise". Now listen to Supreme Leader Snoke's theme from The Force Awakens. You may now shit yourself at the possible implications.
Speaking of "the Tragedy of Darth Plagueis the Wise", the whole story has an ominous feel to it, which is brought to its natural conclusion when Palpatine tells Anakin that Plagueis's apprentice killed him in his sleep. The selling point though is the quiet little smirk Palpatine gives as he muses over the "irony" - it is so clear in that moment that not only is he the apprentice in the story, but that he's recalling and relishing in the exact moment when he killed his master. It's a beautifully subtle look into Darth Sidious's raw sociopathy, and proof he can be just as scary without his face being horribly disfigured.
The horrific ruthlessness of the clone. Surprisingly, according to the monologue in Star Wars: Battlefront II, troopers kind of disliked Ki-Adi-Mundi, but what about Obi-Wan? According to the cartoon, he and Cody were allies and friends for YEARS and he had no hesitation in fulfilling the Order 66. The fact that the clones could be your friends, but be absolute obedient to the point where they will do anything is huge Paranoia Fuel and just plain disturbing.
As explained on the series Nightmare Fuel page, Star Wars: The Clone Wars manages to make Order 66 even more disturbing by giving us more information on how it works, and it turns out the clones are just as much of victims as the Jedi are. The Order is a result of brainwashing chips implanted into them when they're still embryos and are set to activate on a trigger.
Other Expanded Universe material makes it even worse by showing what it actually was: a perfectly reasonable contingence in case the Jedi Order rebelled that they knew about but didn't expect would ever need to be implemented (after all, if the Jedi rebel how do you imprison them? You have to kill them). The worst part is the Irony and Refuge in Audacity of Order 65: it's an order for the clones to arrest the Chanchellor and authorization to kill him if he tries to resist or escape. The brainwashing chip isn't even necessary, in most cases: the Clonetroopers have been just told that a truly nightmarish scenario has started, and they must Shoot the Dog or see the galaxy go to the crapper.
Anakin's implied-but-not-actually-shown murder of the children in the Jedi temple during this scene.
Child: Master Skywalker, there are too many of them. What are we going to do?
His march to the Jedi Temple (With hundreds of Clones in tow) was not much better. While not directly nightmare-inducing, you can tell just by watching (And thanks to Mr. John Williams, hearing) that things are only going to get worse...
In the previous scene, Palpatine ordering the whole thing while grinning like a predator.
And when they finally did operate, Word of God is that it took days, and Anakin was kept conscious for the whole thing. Granted, it wouldn't have hurt quite as much as it sounds; he was burnt badly enough that most of his nerve endings would have been destroyed, leaving him able to feel nothing at all except in a few areas.
If you think the image shown above is scary, than we personally must advise you: don't look up the behind-the-scenes photos of the makeup of Anakin's burn wounds. We're really warning you...
Say what you will about Hayden's performance here, but his screams as Anakin burns to near-death start off as saddened and then just devolves into absolutely blood-curdling.
The look that Anakin makes right before he turns on Obi-Wan. The way it's shot and framed, with Anakin practically 'staring' into the eyes of the audience as he visibly mulls over every perceived feeling of pain, anguish, and utter betrayal.
Consider this: for many children, both Revenge of the Sith and The Empire Strikes Back are probably the first films they've ever seen in which the bad guys actually win; the heroes are either dead or have barely escaped with their lives.
Minor in comparison to the ones above, but in the beginning battle when Obi-Wan and Anakin are fighting their way onto Grievous's command ship. They get attacked by a group of Buzz Droids, tiny droids that like to tear things (including ships and droids) apart. R2 is able to fight them off pretty easily, but R4 ends up getting it's head torn off and sent flying into space squealing all the way. Obi-Wan looks quite perturbed after having witnessed it.
Obi-Wan: Oh, dear...
When Anakin kills Dooku, he's highly conflicted about it. When Vader kills the council in the book, he doesn't bat an eye. In fact, he makes little quips about it. Remarkably funny quips.
The scene is still nightmarish in the film, in part because Vader says nothing as he coldly rampages through the room, butchering the Separatist leaders despite their pleas for mercy and the futile attempts of their Battle Droids to stop him.
The death of General Grievous was pretty graphic for a PG-13-rated film. First, Obi-Wan rips his chest plating open, exposing the working organs suspended in life-preserving gel. After he gets thrown away, Obi-Wan uses his Jedi powers to snatch one of the General's heavy blasters. He fires a shot right into his chest, the General reaching down to grasp at the now burning organ sack, before Obi Wan keeps firing. By the end of his shots, the General has jets of flames shooting out of his eye sockets before he crashes to the deck, every organic part in his body charred to ashes.
And it doesn't help at all that in Star Wars: The Clone Wars Palpatine kidnapped force sensitive infants to said world to perform possibly lethal surgery on them, in a show meant for kids and teens no less.
It definitely doesn't help that in Star Wars Rebels the planet was used for Jedi survivors of Order 66 to torture then kill them. Luminara Unduli was one of these tragic victims.
From the novel, Anakin's Mood-Swinger tendencies get played up, and it's undeniably disturbing, in an alarming realistic fashion. Even Padmé herself, the near single-minded object of Anakin's affections, isn't safe. When she tries to tell Anakin she's pregnant, he instantly becomes afraid she's having an affair, and gets angry, enough that he's hurting her without knowing it.
Palpatine's Force lightning takes a ghastly toll on his own face as he kills Windu. It's right after this that he officially gives Anakin his new Sith moniker: Darth Vader.
Palpatine's demonic snarl he makes while dashing at Mace Windu and the other Jedi masters.