Depending on where you live and what your drainage conditions are like, this can be seasonal or year-round. The itching follows the mold growth. If your house has ever flooded, or if the basement gets wet, you could be in for some nasty black mold called Stachybotros. In quantity, that one can sicken and kill dogs, children and adults, too. Other, less toxic molds, such as Penicillium, are more common.
Molds vary by region, but all can cause allergic reactions in people and animals.
Keep the humidity low in your house by running the air conditioner regularly. Fix any leaks. Make sure the drain is unclogged, and pour a few tablespoons of bleach down the drain pipe every few months to keep it clean. To further reduce mold, install an electrostatic furnace filter and use it. There's usually no need to clean the ducts, unless you've had a severe mold problem, or if the moisture situation hasn't been resolved.