Low Volume Motor Vehicle Manufacturers Act of 2014 - Directs the Secretary of Transportation (DOT) to exempt from certain federal motor vehicle safety and labeling standards (except any standard relating to motor vehicle equipment) up to: (1) 1,000 replica motor vehicles per year manufactured or imported by a low volume manufacturer, and (2) 50 non-replica motor vehicles per year manufactured or imported by a single low volume manufacturer that has evidence of a sales contract.

Defines the term "low volume manufacturer" to mean a motor vehicle manufacturer who annually produces no more than 5,000 motor vehicles worldwide.

Directs the Secretary to permit manufacturers to assign vehicle identification numbers.

Requires manufacturers to register with the Secretary to qualify for an exemption.

Directs the Secretary to require a manufacturer to affix a permanent label to: (1) an exempt non-replica motor vehicle that identifies the motor vehicle safety and labeling standards from which that vehicle is exempt, and (2) an exempt replica motor vehicle that designates the model year that vehicle replicates.

Subjects low-volume manufacturers to all federal motor vehicle safety defect notification, recall, and remedy requirements if a particular motor vehicle has any defects related to motor vehicle safety or any nonconformities regarding any standards other than specified standards.

Amends the Clean Air Act to allow a low volume motor vehicle manufacturer to install in an exempted specifically produced replica or non-replica motor vehicle a motor vehicle engine (including engine emission controls) from a motor vehicle that has been issued a certificate of conformity with Environmental Protection Administration (EPA) emission control standards if certain requirements are met.