Some of the side effects of eating too many cough drops include rapid or shallow breathing, abdominal pain, nausea, vomiting, rapid heart beat or dizziness. While it's possible to experience these symptoms, people are not likely to consume enough cough drops to cause any serious harm.
The primary active ingredient in most medicated cough drops is menthol, which is the ingredient that would cause any negative side effects. It is, however, more likely for people to experience indigestion before any more serious side effects would take place. If someone is deeply concerned about the possibility of over-consumption, it's possible to find cough drops that are not mentholated.