There are two types of tire damage that might be covered:

Slashed or stolen tires

Comprehensive coverage is designed to cover non-collision incidents, like theft or vandalism. If your policy includes comprehensive coverage, then replacement for a slashed or stolen tire may be covered.

Pothole or roadway damage

Collision coverage protects you against crashes on the road, including with another car or object in the road. You might be covered if you hit a pothole or roadway debris with collision coverage.

Flat tires

Flat tires may be covered by roadside assistance. Check your policy to see if it's included. It all boils down to the type of roadside plan you have. Patching a hole from a nail is typically not covered. Luckily, tire repairs are only about $20, according to Consumers' Checkbook.