Financial Institution Customer Protection Act of 2014 - Prohibits a federal banking agency from suggesting, requesting, or ordering a depository institution to terminate a specific customer account, or otherwise restrict or discourage it from entering into or maintaining a banking relationship with a specific customer, unless: (1) the agency has a material reason to do so, and (2) the reason is not based solely on reputation risk.

Prescribes requirements for notice from the federal banking agency to the depository institution about such a customer account restriction or termination; but states that notice to the customer is not required.

Amends the Financial Institutions Reform, Recovery, and Enforcement Act of 1989 to revise requirements for summoning witnesses and requiring production of books or other records the Attorney General deems relevant or material to a civil investigation in contemplation of a civil proceeding which may result in civil penalties for specified violations.