The most obvious to me is a fish.
Menahaden (known as bunkers when I grew up) are inedible. They are filled with tiny bones, with very oily flesh, and they seem to go bad as soon as they are caught.
They are fished commercially, but only as fertilizer, bait, fish oil, and fish meal (for agricultural feed).
Sailors.
Cats are eaten. Douglas Adams’ non-fiction book “Last Chance To See” has a scene that takes place in a market like jovan mentioned, and one of the crew is so distraught by the cats being sold for meat they want to buy them to set them free. In the US we don’t tend to eat non-aquatic carnivores, but that’s just us.
In situations where folks have other choices, do animals like raccoons, members of the weasel family, big cats, wild canines, and bears get eaten? They’re all edible if neccessary, but you don’t hear about fox or ferret being on the menu too often.
!!! In this classic Thomas Nast cartoon, “Coolie, Slave, Pauper, & Rat-Eater” is part of the charges leveled at the Chinese by 19th-Century American xenophobes. Apparently it is not entirely a canard.
[watches thread nervously]
If it moves, they’ll kill it and eat it in Cambodia.
This has come up before, and I’m sure someone told who eats cats, but I can’t remember where that was.
As for your later question about rats, they’re a delicacy among some Thais and elsewhere in Southeast Asia. Never tried them myself. Blecch.
And also in Cambodia is the Dead Fish Tower Inn, whose signs on premises proudly proclaim: “We don’t serve dog, cat, rat, or worm!” There’s a reason they have to emphasize that. 
I’d think that too, but people eat quail, which are no bigger than hamsters.
People eat ants and crickets and shrimp, but by the bunch. People eat quail and guinea pigs individually.
I have to think there’s a category of animals which is too large to be eaten in bunches but too small to eat singly, and that these animals have never been food animals except in emergencies.
Naturally I have no cites for this speculation.
<As for your later question about rats, they’re a delicacy among some Thais and elsewhere in Southeast Asia. Never tried them myself. Blecch>
I lived in Thailand when I was about 9 years old. The things I saw hanging in markets put me right off of meat even at that tender age. Seriously…hanging in a meat market, little bitty skinned things hanging by their hind legs.
The stuff in the outside market, mile long market in particular, covered in triple swarms of flies really finished me off meat for some long time.
Myself, I’ve always found the pig heads hanging from hooks to be unique. A favorite photo op of every tourist who wanders into a market. 
There’s probably no animal that no one has ever eaten, but in terms of economics, it doesn’t make as much sense to raise carnivores for food, because they’re more expensive to feed than herbivores. In order to raise one lion to maturity, you’d have to feed it a whole lot of herbivores, when you could just eat the herbivores yourself.
Of course, this kind of consideration wouldn’t apply when you’re talking about catching wild animals for food; I’m talking about raising them in captivity for the purpose of eating them.
I have no first hand experience, but according to the articles on the page linked to, “Cat meat is popular in the Jiaozhou and Shantou areas of Guangdong province.”
My first thought was great apes. It’s hard to imagine anyone eating a chimp.
Well actually that’s a lie. My FIRST thought was in response to the “cat” comment… I know Mrs. Mooney would have something to say about it.
From my own albeit limited experience in China here “popular” means that you can easily find it if you’re looking for it. However, it’s nowhere near pork or duck as far as popularity goes.
And, I feel this needs to be said, you’re never going to get served dog or cat or even squirrel by accident in a Chinese restaurant. On the contrary, you’re infinitely more likely to get some chicken if you do order squirrel. All “unusual” foods are considered delicacies and they’re much more expensive than more common meats (pork, duck and chicken).
Tadpoles - although frog legs are a delicacy in many cuisines, and I’ve read that toad soup is popular in some SE Asian countries, I’ve never heard of anyone cooking the young of either critter.
Along a similar line, I never heard tell of anyone eating other amphibians: salamanders, newts or mud dogs.
While hungry trappers in frontier days probably chowed down on **wolves, foxes **and various mustelids (weasels, wolverines, fishers and their kin) in times of near-starvation, one never reads of any of them being eaten regularly or indeed at all.
Raptors or birds of prey (taxonomically including big scavenger birds like --yucko!–buzzards) never seem to be on the menu anywhere, either.
I think I’ve read of the flesh and organs of various big cats being consumed for ritual, magical and medicinal purposes, but never as items of cuisine --again, people who were real, real hungry have undoubtedly done so during desperate times, but nobody regularly sits down to a meal of cheetah fajitas or tiger-flank stew at the end of a hard workday as far as I know.
Cat was consumed as a hangover cure in renaissance Spain.
Dammit, I can’t find my cite. But apparently at one point, cats got scarce, and innkeepers started substituting rabbits for cat.
Third or fourth hand…Some travel writer (Tim Cahill?) said that he had once been told, in no uncertain terms, by someone who claimed to have once been so desperate he’d actually done it, that anteater meat was ABSOLUTELY to be avoided. “Rancid” was one of the adjectives used, and not referring to decomposition. I’d hazard that if Colombian jungle outlaws feel that strongly, it’s unlikely that anteaters are in anyone’s recipe book.
Some cultures in Cameroon will gladly eat cat, and I’ve seen it on the menu (as “chat tigre”) at plenty of restaurants. Once, an American friend of mine went on vacation. She gave very stern warnings to her neighbors to stay away from the cat. When she came back, she learned that they had indeed made dinner out of it. But, they said, she shouldn’t be mad because they saved her the pelt.
Porcupine is a pretty common menu item, and was the first meat I ever ate here. Rats are a delicacy and during rat season it’s pretty common to see people walking with sacks full of rats.
Hunting of large primates for food is a good part of why they are endangered. Gorilla meat can be found if you look for it.
So yeah, I’d say there are very few animals that arn’t eaten.