[TripletDaddy]

This is a very basic code section, but I am convinced that most have never read it.

In relevant part:

To be tax-exempt under section 501(c)(3) of the Internal Revenue Code, an organization must be organized and operated exclusively for exempt purposes set forth in section 501(c)(3), and none of its earnings may inure to any private shareholder or individual. In addition, it may not be an action organization, i.e., it may not attempt to influence legislation as a substantial part of its activities and it may not participate in any campaign activity for or against political candidates.

With regards to lobbying specifically:

In general, no organization may qualify for section 501(c)(3) status if a substantial part of its activities is attempting to influence legislation (commonly known as lobbying). A 501(c)(3) organization may engage in some lobbying, but too much lobbying activity risks loss of tax-exempt status.

Legislation includes action by Congress, any state legislature, any local council, or similar governing body, with respect to acts, bills, resolutions, or similar items (such as legislative confirmation of appointive office), or by the public in referendum, ballot initiative, constitutional amendment, or similar procedure. It does not include actions by executive, judicial, or administrative bodies.

An 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.

Organizations may, however, involve themselves in issues of public policy without the activity being considered as lobbying. For example, organizations 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.
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Obviously, that the Church spends a "significant" portion of its time lobbying is not likely. The educational exemption goes out the window when we are told by religious leaders to teach the congregation on Sunday.

Overall, will it hamr their status? Extremely unlikely. But it certainly flirts with the line and is definitely NOT the behavior of a politically neutral organization.