Transit Accessibility Innovation Act of 2014 - Directs the Secretary of Transportation (DOT) to implement a transit accessibility innovation program by distributing competitive discretionary grants to public transit agencies for eligible projects in order to encourage public transit systems to take action to address deficiencies in service for individuals with disabilities.

Directs the Secretary, for each fiscal year, to use 2% of funds made available under this Act to carry out activities to ensure that innovative practices, program models, and new service delivery options are collected, reviewed, and disseminated to other public transit agencies for replication in other communities.

Directs the Secretary, for each fiscal year, also to use certain funds to carry out a transportation promotion pilot program to ensure that: (1) public transit agencies fulfill their requirements under the Americans with Disabilities Act of 1990 (ADA), and (2) individuals with disabilities have advocates to ensure greater opportunities for integration and access into transit systems.

Requires the Secretary to make grants to: (1) agencies implementing a system established under the Developmental Disabilities Assistance and Bill of Rights Act of 2000 that have demonstrated histories of transportation expertise or advocacy, and (2) nonprofit organizations that have also demonstrated such histories.

Increases from 10% to 15% the limitation on the use of a recipient's annual formula apportionment for provision of nonfixed route paratransit transportation services in accordance with the ADA.