Well, the one real objective problem with eating dogs is basically the same as the one that crops up with pigs. Because of co-habitation, dogs have picked up some pretty grungy species-crossing parasites which could be passed on through improperly cooked meat. Dogs and humans have co-existed for a good ten thousand or more years, plenty of time for diseases to jump species. I'm sure any owner of an outdoors puppy has gotten a case of ringworm at some point in their life, or knows how easy it is to pass along. Giardia is another parasite that's pretty common for all sorts of animals to have, and can cause some pretty nasty tummy upsets for humans.
Less objectively, it's difficult because dog faces look a lot like baby human faces in proportion (forward-facing, large eyes, large ears, etc.), so human instinct says it's bad to harm them, just as instinct keeps us, mostly, from hurling babies into walls, no matter how much they scream at nothing. It'd be difficult to find someone to breed and slaughter dogs in the Western world, and harder still to convince people to eat them.
Subjectively, I couldn't conceive eating a dog or a cat. I think we've made dogs at least half-human through breeding and training, and I can't really eat a human. Hell, I think most dogs are more human than most humans are. I know I certainly feel more for dogs that suffer than humans that suffer. Basically, that's the reason I'll eat only either buffalo that I know was raised locally, or chicken. The buffalo thing because I know where they come from, and they do real well by the buggers, and chickens because I raised them, and they're dumb as dirt, even the pet ones.
Anyway, the point is that dogs have something which Ian Stewart and Jack Cohen (of
Science of Discworld fame) refer to as "mind". They describe mind as the part of the brain that is does what we think of as thinking. When you type, that's not mind, but when you learn to type, or develop a new keyboard, you're using mind. Essentially, mind is something you grow over a period of time. Children don't have as much as adults, and, if you're using your brain correctly, you'll have a lot more as an older person than you do when you're 15. They argue that animals we associate with are taught to develop minds of their own, so that a dog won't function (as many dog psychologists will erroneously tell you) mostly on instinct, but more like humans do, with the mind part of them trying to subdue the instinct.
So, in eating something with a mind, you're eating something that you could almost say had a soul. At very least, it's got the rudiments of self-consciousness that we think makes us human and ethical and all that. So, when you eat a dog, or a well-socialized cat, or a pet bird, or a dolphin, an elephant, anything else that has this quality of being a little bit self-aware (beyond reaction to stimuli; elephants can actually recognize themselves in mirrors), you're denegrating that part of yourself that strives to better the world.