About.com has a pretty good primer on the concept of religious privilege:
Religious Privilege as a Nonconscious Ideology:In significant ways, religious privilege is what is known in the social sciences as a “nonconscious ideology.” This means it is perpetuated in ways that people aren’t conscious of. The very environment which structures their lives and assumptions about life also reinforces the assumption that religion is and should be treated with greater deference, respect, and favoritism than other beliefs. Unpacking these assumptions and revealing them is critical to ending religious privilege.
Modern Liberalism vs. Social Privilege:Social, cultural, and political privileges of any sort — gender, race, religion, ethnicity, sexual orientation, etc. — are anathema to modern liberal conception of society. Under modern liberalism, all citizens of society should be treated as fully equal with the same rights, privileges, and responsibilities as everyone else. If one group is privileged over others on the basis of something like race or religion, this institutionalizes discrimination and bigotry.[...]
Religious Organizations & Believers Should be Bound by Neutral Laws & Regulation:The main premise employed by those who advocate religious privilege in modern society is that religion, religious groups, religious beliefs, and religious believers should not have to be bound by the same laws as everyone else. The primary argument against religious privilege should therefore be that neutral laws should be binding on everyone.[...]Defenders of religious privilege tend to frame it as a matter of religious liberty: when laws infringe upon religiously motivated actions, then that infringes upon religious liberty. In the abstract, that sounds reasonable, but when we come to actual cases we can find many instances where one person’s interests may trump another’s free exercise rights. This is an important principle to recognize because too often, people fail to understand or even simply deny the possibility that religion and religiously motivated behavior can cause harm.The right of a person to practice their religion cannot entail a right to harm others or to undermine the public good. The rule of law becomes meaningless if religiously motivated behavior stands outside legal regulation.