Is a borough responsible for damage done to a car tire due to poor maintenance of curbs?

They probably are not liable:

1) First, you need to confirm the borough, not the adjacent home or business owner, is responsible for the curbs in the first place. They most probably are, but it's good to double check--different municipalities can have different rules in regards to maintenance and liability.

2) Second, and more importantly, the rule for municipality/county/state liability for road, etc. maintenance-related damage generally is that only if the government unit had actual notice of a defective condition, would they be liable. So unless there was a complaint about this curb, putting them on notice, they would most likely not be liable.

3) Third, and also critical--cars are not supposed to hit curbs; the curb is not part of the driving surface. Even if you passed the hurdles above, you'd have to show that you were driving/parking in the appropriate place, on the road, when the damaged occured (e.g. there was rebar or the like jutting out from the curb, into the road). If the tire was damaged because you backed into, ran up on, rubbed against, etc. the curb, the damage would likely be attributed to your own driving--i.e., it would be seen as your fault, for driving where you shouldn't, not the borough's.