What happens if too much copper is removed through chelation?  

Several effects. Copper deficiency can cause a number of problems such as neurologic (nerve) problems, anemia (low red blood cell count), confusion, fragile hair, weakness, osteoporosis (thin bones that fracture easily). These effects can be prevented by taking a copper supplement and restoring normal copper balance.
Copper deficiency. So if you decide to chelate, do so under the guidance of a qualified physician and take replacement trace minerals on the off days from the therapy.