That Darn Cat
The Darn Cat from the title is not hurt and does not die.
That Darn Cat!
The Darn Cat from the title is not harmed and does not die.
A pet rat named Beany is exposed to some peril (submerged in a breathable goo, put into an inflated ziploc bag to prevent drowning) and at times it seems like he's gone and has died, but he survives and is not injured.
The Addams Family
Off-screen Grandmama chases a cat and a dog; she implies that she's trying to catch them for dinner. The animals are heard but not seen and we do not see her catch or eat them.
A neighbor's dog comes into the haunted house, sits down staring at a corner, then lays down. Someone checks on it and says it's barely breathing. In the next shot, the dead dog is seen being carried out of the vet, wrapped in a blanket.
A bird is sucked into a device that disintegrates it.
The dog faces a fire but escapes unharmed.
The Autopsy of Jane Doe
Stanly the kitty is found very mutilated in a vent by the son. His neck is then broken by the father in order to put the poor cat out of its misery. The animal is shown dead, but it is ultimately a hallucination.
It's foretold through the storybook that Amelia will kill Bugsy (the dog), Samuel (her son), and then herself. When she is possessed late into the film, she kills Bugsy by snapping his neck. We see his limp body in her hands.
The bear cub that is the focus of the film does not die. His mother dies in a rockslide and another bear is shot and injured but does not die. A bear kills and injures a number of hunting dogs.
A dog and a male ferret both die heroically. A female ferret survives and has a few baby ferrets that also survive.
The Blob (1958)
A dog appears throughout the movie. At one point the main character believes the dog has been eaten by the blob, but she is told by others that they saw it run away. The dog is not seen again.