K. Rool himself from DK64. The ominous music is a bonus, but the voice acting in the intro is a bit of Nightmare Retardant given its similarities to Dr. Claw from Inspector Gadget.
Dogadon in Fungi Forest: after dealing a certain amount of damage to him, he starts making the platform sink, as the music gets faster and faster as the platform gets lower.
The wind-up Kremlings from Frantic Factory are Nightmare Fuel incarnate. Everything about them is unsetting: glowing red eyes, circular walking pattern when not seeing you, squeaky noises when seeing you.
"Creepy Castle" in Donkey Kong 64 had rotting hands as platforms. Inside the actual castle, there's a torture chamber after a dark maze and a very realistic skull-shaped portal leading to a red maze made completely out of bones.
Creepy Castle had an unseen individual moaning constantly.
Angry Aztec is an eerie temple with ominous music that just makes you feel uncomfortable, and once you grab the banana at the end, a voice suddenly shouts "GET OUT!" and aims at you with some sort of gun. You then have 25 seconds to get out of there before being shot down. Said "GET OUT!" comes from an assassin who tries to kill you once you get the bananas. In certain Crystal Caves challenges and in the greenhouse in Creepy Castle, he returns if you fail a challenge and gives you only 10 seconds to escape this time.
One Chunky Kong challenge in Crystal Caves has you avoiding lights. Hit a light, and you get killed instantly after one second!
The Crystal Caves seem like a relatively calm place at first, it's very ambient and you're just walking around gazing at the serenity of the area until a scary noise accompanies stalactites falling everywhere.
Good news: it's possible for Tiny Kong to locate the source of the falling stalactites. The bad news: it's a gigantic Kosha, which is about ten times her size! He still goes down normally, but YEESH!
The fact that the entire Hideout Helm level was timed, considering that if time runs out, all of DK Isles gets destroyed.
The Game Over screen (thankfully skippable, though) has the evil laughter K. Rool showing the Blast-O-Matic about to blow DK Isles up, which was pretty intense for some back when they were children, joining the ranks of Banjo-Kazooie in the "save and quit is not an option" territory.
The Golden Banana where you have to swim into the giant mechanical fish. Not helped by the fact that a late-game glitch renders the banana almost impossible to obtain after getting the Sniper Scope if you don't know how to circumvent it.
Anytime you asked Wrinkly - Cranky's dead wife - for hints, she would always enter with her wailing.
Wrinkly is heavily implied to have died between Donkey Kong Country 3 and DK64.
The constant crying of the Banana Fairy is unnerving as well.
The Evil Laughter that occurs throughout the game is nerve-wracking, especially when you least expect it, like when using a barrel ability or an animal pal ability where it's not allowed.
The Green Klap Traps in Angry Aztec don't die right away. If you defeat them, their teeth survive and come after you.
Fungi Forest at night is pretty scary to those unfamiliar with the level layout.
Creepy Castle's mine cart section. It's bad enough with the loud chase music and random sirens going off constantly, but how about the tombstones that fly out the ground as you approach or the giant skeleton demon thing shooting explosive fireballs at poor DK?
The tunnels in Jungle Japes and Gloomy Galleon have whispers echo throughout.
The secret cavern reached in Jungle Japes by pounding a large X as Chunky has a narrow path above a Bottomless Pit lined with massive torches and a giant creepy face at the end. The music certainly doesn't help matters.
Said creepy music is reused in a beehive within the same level, and the ghost ship in Gloomy Galleon.
As Donkey Kong, you have to complete a few Barrel Cannon challenges that are set in the sky, while a windy and lonely rendition of the aforementioned cave music plays. These challenges tend to invoke feelings of acrophobia.
Here's one for apiphobes (those who fear/hate bees): Zingers are back, and you can shoot them down. Not scary? Well then, try aiming at one at first-person close enough for it to bear down on you. Hope you're quick on the draw.