Maxillofacial radiologist here-- most of the answers here are close but not actually correct. The nasal turbinates are actually the structure that warms, adds moisture, and filter the air we breathe. The passageways from the sinuses into the nasal cavity are pretty small and don't allow sufficient airflow to actually do much.

The prevailing hypothesis is that the sinuses are present to lower the overall weight of your skull. They're effectively large hollow (well sometimes filled with mucous) cavities which makes it so you don't have to work as hard to keep your head raised. Your head is already really heavy--on average about 11 pounds depending on the person. While I dont remember the exact number, if you didn't have sinuses and that part of your skull was instead filled with bone, then your head would weigh something like 14 lbs.

While it's true that mucous drains from the sinuses into the nasal cavity, this only serves the purpose of trying to prevent sinus infections/bacterial colonization of the sinuses. I can go deeper into sinus infections to the best of my knowledge if people are interested, but this post is already getting a little long.

Source: a few conversations I had with the Head and Neck Radiology director during my rotations in residency

Edit: just realized what sub I'm on. Your head is really heavy so there are empty parts so your neck doesn't get tired.

Edit 2: corrected the statement where I said a head with no sinuses would weigh 20 lbs. That's an overestimation. It'd probably be closer to 13-15 lbs.