This question has both a yes and no answer.
From squirrels to deer, if you are hunting native game animals on public land, yes, you should have a license. An airgun hunter has to abide by the same laws that apply to hunters that use firearms.
By law, 100% of the license fees you pay to hunt, fish, or trap goes to the funding of state wildlife resources and conservation programs. This helps to ensure there will be hunting and fishing opportunities for future generations.
Some entrepreneurial landowners have invested in breeding exotic game animals for pay-hunting customers.
Some landowners have even gone all out and offer helicopter hunts, you pay to be flown around and shoot feral hogs from a moving helicopter.
This is done on private land, and most state laws do not apply to exotic game animals or invasive species that are hunted on privately owned land.
Remember, always check the air rifle hunting laws in your area before you go hunting.