From: The Tampa Tribune (Friday, May 31, 1996)

Chlorine in swimming pools can cause health problems, warns a leading pediatrician in a recent issue of Child Magazine.

    “Swimming pools need chlorine** to prevent the spread of harmful bacteria and viruses, but for some kids, exposure to this chemical can worsen or cause health problems in certain areas of the body,” says Jeffrey Brown, also a contributing editor for the magazine.
Pinkeye of conjunctivitis: All children who swim under the water in chlorinated pools are at risk for developing a form of conjunctivitis or pinkeye. “have your child wear goggles if he swims underwater and is old enough to keep them on”, advises Brown.

** NOTE: Actually, no, chlorine is not 'needed', but rather some form of anti-bacterial process is required.

Stomach cramps or diarrhea: This can be caused by accidentally gulping down large quantities of chlorinated water. Children should be taught not to drink or swallow pool water. [NOTE: Chlorinated pool water, that is]
Wheezing: Kids with asthma sometimes find that the chlorine fumes given off by pool water cause them to wheeze. Limit the time that a child spends in the pool if this occurs. [NOTE: If asthma is the cause, then this is not a problem with non-chlorinated ionized water]
Irritated skin: children with eczema, dry skin or rashes may find that chlorine irritates their already inflamed skin. To prevent this, shower or bathe the child immediately following a dip in the pool. 
Sinus congestion: children with nasal allergies can experience a worsening of sinus congestion when chlorinated water rushes up their nose. Have your child try to avoid diving up or doing underwater somersaults and handstands.