Fairness in Flood Mapping Act of 2010 - Requires the Administrator of the Federal Emergency Management Agency (FEMA) to reimburse an eligible entity (i.e., a property owner, a lessee of property, or a community) that incurs expenses for the services of a surveyor, engineer, or similar service provider in obtaining a letter of map amendment that is approved in whole or in part by the Administrator, with respect to the part approved, if the approval was obtained during FY2003 or thereafter.