The core of an apple contains cyanogenic glycosides, better known as cyanide. When your dog munches on these seemingly harmless treats, he also ingests the toxin. Some of the symptoms of toxicity are apprehension, dilated pupils, salivation, struggling to breathe, dizziness, collapse, seizures, hyperventilation, shock, and coma.
Along with a few other fruits (plums, peaches, pears, and apricots) you should definitely be careful not to leave apple cores laying around for dogs to get their paws on.