Benji: The Hunted - Benji, a dog, adopts some orphaned cougar cubs. He stays out in the woods to raise them even though he wants to go home to his master. One scene gets me every time: A hawk picks up one of the cubs, and you can hear it screaming as it's carried off. At that point I turned the damn movie off. I've never seen the ending.
The Brave Little Toaster - I'm pretty sure the movie is a metaphor for people who want to make their situations better, and how some people want to live better but they can't because they're disabled or sick and have no means to climb up. (I looked it up after typing this, and I was correct! Each character is a metaphor for a type of person.) This movie upset me too much to watch even when I was little.
The Plague Dogs - Two dogs escape from an animal testing lab. One dog was involved in drowning experiments and is afraid of water. The other dog had brain surgery and keeps having seizures and hallucinations. The animal testing was enough to freak me out, but the dog afraid of water did everything he could to protect the smaller dog. They try to survive on their own while the scientists from the lab try to track them down. If you need further information on how horrifying this movie is, it was written by the same guy who wrote Watership Down. Think about that.