TL;DR version: There seems to be some argument about whether dogs are carnivores or omnivores. It is more important that you follow the guidelines for how much meat and plant matter to feed your dog than to worry about the taxonomic distinction that dogs fall into.
Case for omnivores
Humans and dogs (and pigs and bears) are monogastric omnivores. This simply means that they have one stomach [...] and consume a variety of animal and plant items. Pigs may eat far more vegetables and dogs much more meat, but we all find ourselves somewhere along the same continuum.
Cats, on the other hand, are obligate carnivores. While dogs can survive (though perhaps not thrive) on a vegetarian diet, cats must eat meat to fulfil their nutritional requirements. [Goes on to describe how dogs' teeth and taste buds are useful for eating vegetable matter.]
Source: Grab Life by the Leash
A dog's diet should consist of balanced porportions of proteins, carbohydrates, fats and, of course, water. A dog can go days without food and lose 30% to 40% of it's body weight without dying, but a 10% to 15% water loss could be fatal. All-meat diets are not recommended for dogs due to the lack of calcium and iron found in meat. Diet supplements should be avoided. Human foods that can be fatal to dogs include moldy cheese, onions, and chocolate. Feral domestic dogs will eat a variety of foods including animals and fruits.
Primary Diet: omnivore
Source: University of Michigan Museum of Zoology
Case for carnivores
Dogs do not normally produce the necessary enzymes in their saliva (amylase, for example) to start the break-down of carbohydrates and starches [...] Nor do dogs have the kinds of friendly bacteria that break down cellulose and starch for them. As a result, most of the nutrients contained in plant matter—even preprocessed plant matter—are unavailable to dogs.
Source (author doesn't really cite her experience - seems to be a personal website, albeit one with a lot of quotes)
With regards to keeping a dog as a pet, I'm not sure that the actual distinction matters. (If you want a deeper knowledge than that required to keep a dog as a pet, the Biology Stack Exchange should have plenty of experts who might be able to give a better answer than us.) Further, there is a large variation in how much meat is eaten within both the "carnivore" and "omnivore" spectrums.
The key things to note are:
- Dogs, like us, do not thrive on a vegetarian diet.
- They also do not do well on an all-meat diet.
- It would be best to feed your dog a balanced diet that contains both animal and plant items.
- Different dogs require different levels of meat in their diets. Chihuahuas (and many other toy breeds) have evolved to eat predominantly a "people diet" of vegetables, corn and only a little meat, whereas war dogs like mastiffs were bred an an all-meat diet. Source.
How will not feeding them meat affect their health?
[T]he protein and calcium needs of the dog and cat are much higher than those for humans. These nutrients are most easily provided through animal-derived ingredients. Some plants, such as soy, are high in protein, but the amino acids within the protein are not as balanced as they are for most animal-source ingredients. Dogs and cats also need a dietary source of vitamin B12, a substance not found in most plants. All animals "need" this vitamin, but plant-eating animals such as cattle and sheep can make their own through the action of bacteria in their gastrointestinal tracts, provided there are adequate amounts of the mineral cobalt in the diet (which is found in plants).
How will not feeding them plants affect their health?
There's a distinction between different types of carnivores:
Animals that depend solely on animal flesh for their nutrient requirements are considered obligate carnivores while those that also consume non-animal food are considered facultative carnivores.
Source: Mammals: Carnivores. Duane E. Ullrey. Encyclopedia of Animal Science. - reference in Wikipedia
Even those sources that claim that dogs are carnivores are not claiming that dogs are obligate carnivores. In other words, even if they are carnivores, they are not the type of carnivore that eats only meat.
[T]here is no such a thing as an all-meat diet in the wild. Wild wolves inevitably digest a fair amount of plant matter too – as they benefit from the plant diet of the prey animals they hunt.
And finally, as quoted above:
All-meat diets are not recommended for dogs due to the lack of calcium and iron found in meat.