If you give too much insulin relative to the body's need, blood sugar can drop too low, and that would be if you've been more active and burnt extra sugar or haven't eaten as much. Now, for most low sugar reactions, they're relatively minor, can cause some uncomfortable symptoms. On rare occasions, particularly with type 1 diabetes, if you really overdo it, the patient can have mental status changes or even become comatose and in extremely rare circumstances could even die. It's important to get a proper balance between the insulin and the food and the activity. The other side effect is weight gain.
People are very concerned that insulin can cause weight gain.
Now, insulin itself, of course, doesn't have any calories. It's not the insulin that makes you gain weight but rather the imbalance of the insulin with extra food. I mentioned a moment ago it can cause the sugar to be low. You take more food. And if this is happening fairly frequently and stimulating the appetite, people are going to double the amount of calories they take in potentially, and thus gain weight.