Is it okay to use fine thread drywall screws when securing drywall to wood studs? What’s the worst that could happen if not?
It's fine. They are primarily intended for metal studs but they will work fine in wood. Think about the force required to physically pull out one screw - even a fine threaded screw. Now given the fact that each sheet of drywall will typically have about 32 screws, the drywall itself will fail long before the screw is pulled out. To make yourself feel better, drive a fine threaded screw 3/4 of its length into a stud. Then grab pliers and try to pull it out...
Don't use fine thread. Use the coarse thread screws. Over time wood expands and contracts with heat and humidity. Combine that with the weight of the drywall they WILL pop eventually. The question is will you still be there to fix all the popped screws? There are different threads for a reason if your going to spend the time doing something do it right the first time.