They do. Dogs and cats frequently suffer from a buildup of tartar, which accumulates like cement on the teeth and causes gum irritation. If left long enough, eventually the gum regresses so far the tooth roots are exposed and the tooth has to be removed.
Animals can also have fractured teeth - owners often don't notice them, because if the animal has had it for a while the tooth may be dead and won't hurt - but you know yourself that a cracked tooth is initially very painful!
Tooth root abscesses are common in rabbits, cattle, horses, and hamsters. They are not so common in cats and dogs but we still see themfairly often. Best case scenario - the breath stinks but there is no lasting damage and a short course of antibiotics clears it up. Worst case scenario - infection has established itself in the bone (very painful) and there may be facial swelling or the eye may stick out more - and it may then be too late to treat effectively.
Animals can't tell us they have toothache - but you can tell if their mouth is sore by watching them eat. They may eat slowly and careful, or they may refuse dry food and only lick jelly off wet food.
Our sugary diets increase our chances of tooth decay, but our animals don't escale the dental disease either!