Artificial food dyes have been around for years, and many forms are actually approved by the U.S. Food and Drug Administration (FDA). Recently, there have been more studies linking food coloring with hyperactivity in children. The FDA is currently reviewing food dyes and considering banning some.
There is no conclusive evidence that food coloring causes nor increases hyperactivity in children. There is, however, some indication that children with Attention Deficit Hyperactivity Disorder (ADHD) may be more sensitive to dyes that other children.
For more information read up on the FDA's released analysis of 35 years of scientific studies.
My two cents? An elimination diet, when it is not too restrictive nor severe, is never harmful. Use natural products and avoid colored drinks and food when possible. Keep an eye on your child's behavior during this time. Whether or not you see a difference, you would have introduced a healthier diet to your child :)