My pediatrician says I don't need to worry about my children between the ages of 2 years and 18 years old having iron overload. He says it's an "adult onset" disease; is this true?

Worry about iron. You must realize that your doctor is talking about safe nutritional doses of iron for the kids. One of the common poisoning that occurs in children is iron, usually in the form of candy like multivitamins which are now phased out. Growing children and menstruating women need adequate iron on a daily basis. Other adults do not usually need much iron unless they are anemic due to iron deficieny.
Not entirely true. Children with hemolytic disease, e.g., sickle cell disease, are at risk to develop iron overload when they receive frequent transfusions or iron supplement.