Comprehensive TB Elimination Act of 2014 - Amends the Federal Food, Drug, and Cosmetic Act to require the Drug Shortages Strategic Plan to include plans to ensure coordination between the Drug Shortages Task Force and the Federal Tuberculosis Task Force.

Amends the Public Health Service Act to require the Department of Health and Human Services, when awarding grants for the prevention, control, and elimination of tuberculosis, to give priority to state health departments proposing to focus on high-risk populations, including foreign-born, homeless, and uninsured populations.

Authorizes the Health Resources and Services Administration to award grants to state and local governments and federally qualified health centers to coordinate their programs and services to ensure timely and appropriate prevention, diagnosis, and treatment of tuberculosis.

Requires the Federal Tuberculosis Task Force to advise federal officials on strategies and systems to prevent and mitigate shortages of tuberculosis drugs.

Reauthorizes the national strategy for combating and eliminating tuberculosis through FY2019.

Allows the National Institutes of Health to enhance research on pediatric tuberculosis and tuberculosis infection and progression.