Two major cases come to mind. Scientology is probably the most well known. It had persistent battles with the IRS over it's tax status. There is some argument to be made that Scientology moved to label itself as a religion principally to avoid taxes. In the early 1950s operating as the Hubbard Association of Scientologists it portrayed itself as either a branch of psychology or a rival to it, not a religious community.
Some of the religious practices that Scientology adopted were specifically to make it seem overtly religious, to qualify for a tax exemption. The Scientology Cross was created to make Scientology seem more like a Christian group . The use of clerical collars and the usage of the title "minister"may have been introduced in the late 1950s for the same reasons. Even the fact Scientologists preformed religious ceremonies like funerals may partly have been an effort to seem more religious to be able to secure tax exempt status.
Scientology actually infiltrated the United States government in the 1970s, partly out of a desire to subvert the IRS in what they called Operation Snow White. They were eventually stopped by the FBI. Scientology was also extremely litigious, repeatedly suing the IRS. Scientology and the IRS came to an accord in 1993, with Scientology agreeing to pay over $12 million in taxes in exchange for the IRS granting them tax exempt status.
The other big example of a religious organization being striped of tax status was Bob Jones University. Bob Jones University is an evangelical school, and at the time it had a policy prohibiting interracial dating or marriage among its students. It was tax exempt as an educational institution and a religious institution, but the federal government argued that being racially discriminatory should cause it to lose that tax exemption.
In the 1983 Supreme Court case Bob Jones University v. United States the university lost. The court found that it did not interfere with the free exercise of religion to remove a tax exemption from an educational institution that was engaged in racial discrimination. Bob Jones was forced to pay back taxes, and kept it's policy in place until the the year 2000.
Recommended Readings:
Dalhouse, Mark Taylor. An Island in the Lake of Fire: Bob Jones University, Fundamentalism, and the Separatist Movement. University of Georgia Press, 1996.
Greenawalt, Kent. Religion and the Constitution Volume II: Establishment and Fairness. Princeton, N.J: Princeton University Press, 2008.
Urban, Hugh. The Church of Scientology: A History of a New Religion. Princeton: Princeton University Press, 2013.
Wright, Lawrence. Going Clear: Scientology, Hollywood, and the Prison of Belief. New York: Knopf Doubleday Publishing Group, 2013.