From the AVIS FAQs:

Country = "USA"
Find a Topic = "Customer Service"
Customer Service, FAQs = "I have vehicle or tire damage, what do I do?"

"You are responsible for any and all loss of or damage to the car resulting from any cause including and not limited to collision, rollover, theft, vandalism, seizure, fire, flood, hail or other acts of Nature. Please call the Avis Emergency Road Service at 800-354-2847 for assistance."

Depending on your insurance and deductible situation, you will probably receive a bill for either repairing or replacing the tire. I have found no benefit to AVIS First; the vehicles are often filthy, poorly maintained, and have 25,000 or more miles on the odometer. I have been renting primarily out of SFO but experienced the same earlier this year at LGA.