A person is not allowed to stand idly by and watch further harm come to his/her property. Everyone must use reasonable care and diligence to minimize the damages - to do what s/he can to prevent additional damage. An injured party must make a reasonable effort or expend reasonable costs to prevent the amount of loss or harm from increasing. For example, if constant rain is raising the level of a lake next to your home, you have the responsibility to take steps to prevent damage from the flood which is about to occur, such as moving property to higher ground or putting out sand bags to prevent water from entering your home. The particular acts required by a person to mitigate damages is determined on a case-by-case basis. It is set as what is "reasonable" under the particular circumstances.
If you are seeking recovery for property damage from an insurance company, you should seek the advice of an experienced insurance lawyer. If the property damage was due to a car accident, see a car accident lawyer right away.