Rule of Personification Conservation

If a work features a non-human character or a non-human cast, then they are non-human for a reason.

Another specific application of Law of Conservation of Detail, saying that the use of non-human characters must be justified.

If your character is a dog, for instance, you might do well to have the story be about his being man's best friend while making references to other dog stereotypes.

If your cast is a perfectly normal family doing perfectly normal things, except they happen to all be anthropomorphic spoons or something, some audiences are going to wonder what the point is.

There are reasons for a cartoon to use anthropomorphic animals instead of humans:
 * The show is being geared towards children, for whom animal characters form an automatic shorthand for child-friendliness. So much so, that if the show isn't for children, people will ask "What Do You Mean Its Not for Kids"
 * Trying to use humans in CGI animation is likely to dig up the Uncanny Valley. Early CGI-animated movies avoided human characters as much as possible (although they did make sure to use plots that had other reasons for nonhuman leads).
 * Even in 2D animation, some people believe that Furries Are Easier to Draw.
 * The creator, scriptwriter, or artist is secretly in the Furry Fandom. (Or, in some cases, openly.)

Consider the Sliding Scale of Anthropomorphism. If a character is a realistic animal, Nearly Normal Animal, or Talking Animal, then it is easy to justify its animal state; you don't want a human to play a perfectly normal dog any more than vice versa. But if a character is a Funny Animal or a Humanoid Animal, then you may need to throw in some deliberate references or allusions to help justify it.

Also consider the Rule of Animation Conservation, another good reason for a filmmaker to use an animal cast- namely, to justify the use of animation. This is circular, natch -- each justifies the other.

Anime

 * So far averted with Sajin Komamura from Bleach who is an anthropomorhic fox humanoid for some unexplained reason. Also done to a lesser extent with Yoruichi Shihōin who turns into a cat but nobody ever really questions this.

Fanfic

 * Transformers fiction sometimes stumbles over this. You can write mundane war stories about Transformers, but if your characters are all alien shapeshifting robots, maybe that should figure into your plot?

Newspaper Comics

 * Gary Larson's The Far Side is a perfect example of this trope in action. Many times, his one-panel strips would employ a cast of non-human anthropomorphic characters doing human things (For example, a family of dogs or even a boomerang husband and wife). When this happened, you could guess from the start that the punchline would have to do with them being non-human creatures (for example, the joke for the boomerang couple was based on the idea that boomerangs come back to you when you throw them).
 * This troper's favorite is one where a mother camel scolds her son for having a glass of water at night because he just had one a week ago.

Video Games

 * The Star Fox games also avert this trope. The characters in this game include an anthropomorphic toad, bird, wolf, hare, and- of course- fox, and they all partake in flying in mercenary aircraft through space. Nothing related to their Species Surnames to this troper's recollection

Western Animation

 * Mickey Mouse. Yeah, he's a mouse and his name's Mickey Mouse, but other than that, there's not much mouse-like about him; almost everything he does seems like something we would do. His biggest enemy is an anthropomorphic cat, but Pete's feline nature isn't always self-evident. But there is a reason for his being a mouse: Furries Are Easier to Draw, especially in cheap black and white animation (how he started), and the species-less shape was being used by someone else. (Back in the late 1920s, many cartoon characters were similar-looking except for the ears and Tertiary Sexual Characteristics.)
 * Wait. Pete's a cat?!
 * Furries are easier to draw, but making it clear what species of furry they are isn't, apparently.
 * Bugs Bunny uses this rule. Most of the things he does are strictly human traits, but the guy lives in a rabbit hole, digs long tunnels (occasionally making the Wrong Turn At Albuquerque), munches on carrots all the time, and constantly has to worry about rabbit and wabbit hunters.
 * Chuck Jones, director of many of the greatest Warner Bros. cartoon shorts, claims in his book Chuck Amuck that the reason animators like to use animals is that "it's easier to humanize animals than it is to humanize humans."
 * In general, Pixar's the king of this trope. The fact that the Cars series is generally considered their weakest work probably has a lot to do with the fact they're the only films where the story doesn't exactly require the characters be cars, and the fact that there are no humans in it, whereas Toy Story is about problems toys would have to face if they were sentient, Ratatouille with rats, Wall E with robots, etc.
 * Kung Fu Panda uses an all animal-cast for the Rule of Symbolism. All the members of the Furious Five use martial arts moves based on their animal namesake. Then there's Po, who, like a panda, is big, lazy, and probably naive about sex.
 * Brian from Family Guy is the Griffin's Intellectual Animal pet dog. He talks, he smokes, he drinks, he argues for the liberal camp, and he kinda looks like a more cynical Snoopy. But there are many running gags based on the fact that he is, indeed, a dog, such as how he likes to chase his tail and sniff other dogs' butts.
 * Disney's Robin Hood deliberately chose to use animals -- it's right in the opening narration. This is partly for the Rule of Symbolism: Robin is a fox because he's clever, Maid Marian is a fox because Robin is, Friar Tuck is a stubborn and tough badger, Prince John and King Richard are royal lions, Alan A Dale is a chanticleer -- um, a rooster... Another reason: this was made directly after the Disney version of The Jungle Book, which meant that half the character designs could be copy-pasted from there. Little John is related to Baloo, and Sir Hiss is related to Kaa... The characters do sometimes slip into animalistic traits. Most bird characters do fly, a young turtle hides his head in his shell, and it's amazing how much mileage one can get out of compressing snakes into small spaces...
 * The main cast of Felidae consists of talking cats. They mostly just talk to one another, but ). They read, they have their own religion (that is, The Claudandus Sect), and one of them is able to use a computer... but they still do normal cat things like hiss, have random sex, and, in the case of Bluebeard, urinate on things to mark their territory.
 * Futurama has Bender, basically an inversion of Asimov's laws of robotics.
 * In Adventure Time, there are lots of characters who are practically human, usually humanoid beings with green or blue skin and maybe some other differences. But Finn is the only true human, and it's been made clear that he's seemingly The Last of His Kind.