When does the home insurance pay to repair a water leak around a brick fireplace chimney?

There are several different types of policy forms used by property insurance companies. For an owner occupied single family home (versus a rental property, secondary residence, or condo) the policy form typically used is called the Homeowners Form 3 (HO-3). The HO-3 policy will cover "sudden and accidental" water damage. What this means is that if you have a constant leak that was in plain view then the damage would not be covered. It has to be "sudden and accidental".

The best way to determine whether or not you have coverage for this is by looking at your contract. The HO-1 and HO-2 policy forms are more limited. In the HO-3 you will want to read everything under the Section 1: Dwelling Coverages, AND the exclusions for Section 1: Dwelling.

The reason that you need to refer to the Dwelling Coverages (Coverage "A" on most policies) is because a brick fireplace is a permanent part of the structure of your home.