No Health Care Subsidies for Illegal Aliens Act of 2010 - Amends the Patient Protection and Affordable Care Act, with respect to procedures for determining eligibility for participation in a state health care insurance exchange (Exchange), to: (1) require an applicant for enrollment in a qualified health plan (enrollee) to appear in person at an Exchange and submit a sworn statement, under penalty of perjury, that the enrollee is a citizen or national of the United States or an eligible alien; (2) require enrollees to provide satisfactory documentary evidence of citizenship or nationality or satisfactory immigration status; (3) require Exchanges to verify citizenship or immigration status of enrollees based on satisfactory documentary evidence and (4) eliminate the authority of the Secretary of Health and Human Services (HHS) to modify the methods used by Exchanges to verify enrollee eligibility.