Hi,
It can be due to the gas or the ice itself. Some people do exhibit such problems and its best that they visit a allergy specialist who can actually trace out the reason for the allergy.
You may in the meantime, protect your child from the ice rink as well try steam inhalation, OTC anti-allergics.
Let su know about your childs health and what your doctor says.
Best regards.
I have asthma and I ice skate. My doctor said that it's very important that I'm not out on the ice for at least 20 minutes after the zamboni leaves becuase the cold temperature of the ice traps the fumes in for longer. Hope that helps.