Can forced air heaters trigger allergies?

Yes: Many people who have allergies are allergic to house dust. Running the forced hot air can circulate this in the house. Using a good filter in the heating system and using an air purifier in the child's room might help reduce some of the exposure.

No: No, but as a simple precaution you should change your air filters at least every 3 months. If your child develops dust mite allergy, then dirty air vents can be a trigger of allergy symptoms, but forced air heaters in themselves will not trigger allergy per se.