A non-profit or 501(c)(3) organization is granted its status by both state and local governments and therefore, may not discriminate in hiring for any unlawful purpose. Discrimination on the basis of religion is illegal. However, being gay/lesbian is not a protected class. Race, age, gender, disability, religion, nationality, color and creed are all protected classes.
Because religion is an illegal basis upon which to make a hiring decision, the not being a Christian would be an illegal basis to discriminate.
My comments are not intended to establish an attorney-client relationship, are not confidential, and are not intended to constitute legal advice. Proper legal advice can only be given by an attorney who agrees to represent you, who reviews the facts of your specific case, who does not have a conflict of interest preventing the representation, and who is licensed to practice in the appropriate jurisdiction where the legal issue may be filed or in the state where the law applies.
This is a difficult question to answer with certainty because there is an exemption from Title VII of the Civil Rights Act of 1964, for some religious groups to discriminate in hiring. Section 702 states, in part, that the following groups are exempt: "a religious corporation, association, or society with respect to the employment of individuals of a particular religion to perform work connected with the carrying on by such corporation, association, or society of its religious activities or to an educational institution with respect to the employment of individuals to perform work connected with the educational activities of such institution." So if the organization you described is hiring for a position that would fall under this category listed in the exemption then they may do so. As for discrimination based being gay, there is no federal prohibition against this, but there may be state or local laws related to this.