The IRS Website lists the ways in which a person can be exempt from the individual mandate. From the website:
- Religious conscience. You are a member of a religious sect that is recognized as conscientiously opposed to accepting any insurance benefits. The Social Security Administration administers the process for recognizing these sects according to the criteria in the law.
- Health care sharing ministry. You are a member of a recognized health care sharing ministry.
- Indian tribes. You are (1) a member of a federally recognized Indian tribe or (2) an individual eligible for services through an Indian care provider.
- Income below the income tax return filing requirement. Your income is below the minimum threshold for filing a tax return. The requirement to file a federal tax return depends on your filing status, age and types and amounts of income. To find out if you are required to file a federal tax return, use the IRS Interactive Tax Assistant (ITA).
- Short coverage gap. You went without coverage for less than three consecutive months during the year. For more information, see question 22.
- Hardship. You have suffered a hardship that makes you unable to obtain coverage, as defined in final regulations issued by the Department of Health and Human Services. See question 21 for more information on claiming hardship exemptions..
- Affordability. You can’t afford coverage because the minimum amount you must pay for the premiums is more than eight percent of your household income.
- Incarceration. You are in a jail, prison, or similar penal institution or correctional facility after the disposition of charges against you.
- Not lawfully present. You are not a U.S. citizen, a U.S. national or an alien lawfully present in the U.S.
So, you can become exempt from the individual mandate by impoverishing yourself, getting yourself thrown in prison, renouncing your citizenship, etc., but you were particularly interested in the religious conscience exemption. The Social Security Administration has the power to determine what religious groups are exempt, and you can apply for a religious exemption using this form.
The form says that approved religious sects are described in section 1402(g)(1) of the Internal Revenue Code, which I managed to find here. To qualify for a religious exemption, you have to show:
(A) such evidence of such individual’s membership in, and adherence to the tenets or teachings of, the sect or division thereof as the Secretary may require for purposes of determining such individual’s compliance with the preceding sentence, and
(B) his waiver of all benefits and other payments under titles II and XVIII of the Social Security Act on the basis of his wages and self-employment income as well as all such benefits and other payments to him on the basis of the wages and self-employment income of any other person,
and only if the Commissioner of Social Security finds that—
(C) such sect or division thereof has the established tenets or teachings referred to in the preceding sentence,
(D) it is the practice, and has been for a period of time which he deems to be substantial, for members of such sect or division thereof to make provision for their dependent members which in his judgment is reasonable in view of their general level of living, and
(E) such sect or division thereof has been in existence at all times since December 31, 1950.
The criteria for religious exemption from the individual mandate and Social Security seem to be the same (as all sources I can find point to the same criteria for religious exemption), though policy may come to differ as people apply for religious exemptions. However, at the moment, it looks like you have to be Amish to qualify for a religious exemption, as they are the only religious group whom I could find that qualify for an exemption from the individual mandate and Social Security.