Harry and Tonto
At the end of the film the cat (the titular Tonto) is euthanized due to illness and old age.
Harry Potter and the Prisoner of Azkaban
Buckbeak, the hippogriff (a mythological creature that looks like a mix of a bird and a horse), a dear pet of Hagrid, is about to be executed by decapitation, and we hear a hacking sound, implying his off-screen death. However, we later learn that he was rescued just before the execution and it was a pumpkin that was hacked at. Buckbeak however eats several already dead ferrets throughout and in one scene catches a bat in its flight and eats it.
A werewolf and a man that has transformed into a dog heavily fight but they both survive relatively unharmed (both of these characters are good). A rat is missing for much of the movie and is thought to have been eaten by a cat bit re-appears later in the movie.
After attacking a man, a dog is punched and then dies after being impaled on farm equipment.
Hector and the Search for Happiness
It appears in the first few moments in the movie that a dog might die, but it isn't so.
The dog is part of his subconscious that is the representation of his happiness. It appears throughout the movie. The movie ends happy with a happy dog.
A large number of cats are seen in Hellboy's home, none of which come to any harm. In a scene set in a subway, Hellboy juggles a box of kittens while fighting an enemy. The kittens are saved and are returned to their owner.
Herb & Dorothy
Documentary. We do not see any harm come to the various cats (including shelter cats), turtles, fish and at least one dog that appear in the film.
Hidalgo is injured when he falls in a pit trap and is impaled on a piece of bamboo but he survives. A different horse in the race is severely injured and is stabbed to death to put it out of its misery.
Hide and Seek
A dead cat is pulled out of a bathtub full of blood.
A small dog is not being controlled with a leash as part of a plot point. The pup is standing in the middle of the highway and is run over by a speeding car. The 3 characters that witnessed Tiger's death do not have much remorse.
In the opening sequence, a puppy is shown suspended in a Christmas Stocking.
A boy who is magically transformed in to an immortal cat is run over by a bus but survives. Later the cat appears to die but it is actually the boy's spirit being released in to the afterlife.
An invisible dog is killed by being smashed against the walls of a kennel. The scene is filmed in heat vision so, despite being invisible, the dog's death is seen.
A kitten is kidnapped by thugs but rescued with no harm done.
Homeward Bound: The Incredible Journey
Two dogs and a cat end up in perilous situations while trying to get home. There are moments when it appears one or another of the animals has died, but in the end they all survive.
Hondo's dog Sam dies after being speared.
Nana, a sheepdog, shows up in a few scenes and is not harmed.