Note to Moderators: I have quoted from the IRS code below - I believe it is free use re: copyright as it is a gov publication.
There is a connection to the 1st Amendment but it is not the only basis for tax exempt status, the other being non-profit basis. The below is a link to the IRS publication (in PDF) that describes the church exemption (why, how to get it, how to lose it).
irs.gov/pub/irs-pdf/p1828.pdf an excerpt:
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A church or religious organization will be regarded as attempting to influence legislation if it contacts, or urges the public to contact, members or employees of a legislative
body for the purpose of proposing, supporting, or opposing legislation, or if the organization advocates the adoption or rejection of legislation. Churches and religious organizations may, however, involve themselves in issues of public policy without the activity being considered as lobbying. For example, churches may conduct educational meetings, prepare and distribute educational materials, or otherwise consider
public policy issues in an educational manner without jeopardizing their tax-exempt status.* … there is quite a bit more, please use link to see original in its entirety.
I see from googling the issue that the Mormon church may face a challenge based on the huge amount of funding it provided to the Yes on Prop 8 campaign here in Calif. There is a line and they may well have crossed it. See below, again from the IRS code:
Under the Internal Revenue Code, all IRC section 501(c)(3) organizations, including churches and religious organizations, are absolutely prohibited from directly or indirectly participating in, or intervening in, any political campaign on behalf of (or in opposition to) any candidate for elective public office. Contributions to political campaign funds or public statements of position (verbal or written) made by or on behalf of the organization in favor of or in opposition to any candidate for public office clearly violate the prohibition against political campaign activity. Violation of this prohibition may result in denial or revocation of tax-exempt status and the imposition of certain excise tax.
Though relgious leaders are permitted to make remarks as individuals, the IRS points out that in order for the institution to remain tax exempt that religious leaders cannot make partisan comments in official church publications or at official church functions.
This is the law as it currently stands, straight from the horse’s mouth. Make of it what you will.