Where I live, fleas are a year round problem. Other than a lot of dogs (and cats) being allergic to flea bites and the resulting problems from scratching, fleas can carry the plague and tapeworm. Fleaborne diseases of the United States | CDC
Flea anemia is also a possibility with severe infestations on very young or very small dogs, though it's more common in cats. That shouldn't be a worry if you stay on top of things and prevent big infestations by treating for fleas when you first notice them, and never let your pets get overrun.
It's less common to find fleas outside during very cold seasons, but they can absolutely infest indoor spaces year round, even in cold climates. In warmer climates, they're absolutely a year round problem indoors and out.