The teenage boy in Asterix and the Normans being surrounded by vicious looking Vikings, until their leader walks towards him and demands: "Make us scared!"
Asterix and Obelix lost at sea to the point of starvation in Asterix and the Great Crossing.
The start of Asterix and the Soothsayer. A storm is raining down on the village and a strange, creepy old man in a wolf skin enters the chieftain's home to predict the future by cutting open a fish. The soothsayer's face can also be seen in a lot of dramatic close-ups in the shadows.
There's definitely a certain atmosphere of Adult Fear in "Asterix and Son" with armed forces kidnapping a baby.
The animated films:
The storm scene in Asterix Versus Caesar (the animated movie). Having Asterix nearly drowning in a dungeon, Dogmatix being flushed through the sewers, and Obelix desperately searching for both of them makes this easily the darkest moment in all history of Asterix.
The soothsayer in Asterix and the Big Fight, especially the way he cuts up a fish to tell fate from it... and then uses it as a hand puppet. It should be funny. It's not.
The Twelve Tasks of Asterix is full of examples.
The Place That Sends You Mad (and an honorable mention goes to the Cave of the Beast beforehand). The House itself isn't that terrifying, but seeing the effects is, especially when there are some genuinely terrifying things among all of the kid-friendly crazies. Woman running down the street acting like a chicken? Hah—man, that's fun. Man with a Slasher Smile, following her with an axe and chopping wildly? Oh... God.
The Egyptian hypnotist is also creepy, with his glowing eyes.
Speaking of eyes, the Roman who ends up giving them permit A38. Think Judge Doom.
In the eleventh task, where they have to sleep on a haunted plain. Like the Cave of the Beast, it's pretty disturbing until the moment Asterix starts to argue with the centurion ghost. At which point it turns into a Funny Moment.
In one scene, Caesar pulls a genuinely creepy Slasher Smile followed by an Evil Laugh, after pondering whether Asterix and Obelix will survive the next task. This is immediately followed by a shot of the senators trying to hide under the table, being just as creeped out as the viewers, brilliantly lampshading this, but not quite managing to serve as Nightmare Retardant.