Could the IRS Deny Tax-Exempt Status to Religious Schools That Discriminate on the Basis of Sexual Orientation?

No, not at this time. In my opinion, the courts would allow the Internal Revenue Service to deny tax-exempt status to religious schools that discriminate on the basis of sexual orientation only if Congress made such discrimination in education illegal. If Congress does that, then of course no school that engages in acts of illegal discrimination should be entitled to claim tax-exempt status.

In The Hill, Bernie Becker reports that religious schools and universities are concerned that they will lose their tax-exempt status if they discriminate against gay and lesbian students or employees. Colleges Opposed to Same-Sex Marriage Fear IRS Retribution (August 8, 2015). Becker quotes Marcus Owens, a former senior I.R.S. official, and John Colombo, an Illinois law professor, as stating that this was not at all likely. Here's why.

Four decades ago Goldsboro Christian Schools of North Carolina refused to admit black children, and Bob Jones University of South Carolina refused to permit interracial dating or recognize interracial marriages. As a result the I.R.S. stripped those schools of their tax-exempt status, reasoning that they were not "charitable institutions" within the meaning of Section 501(c)(3) of the Internal Revenue Code. In the case of Bob Jones University v. United States (1983) the United States Supreme Court upheld the decision of the I.R.S., reasoning that  racial discrimination in education is so contrary to law and public policy that it was appropriate for the I.R.S. to find that schools that engaged in that practice did not qualify as charitable institutions. The schools argued that donations to their institutions ought to be treated as tax-deductible, but the Supreme Court cited and quoted an old case, Perin v. Carey (1861), where the Supreme Court had added a "caveat" that the courts would uphold

a gift ... to public charitable uses, provided the same is consistent with local laws and public policy ...

and that it was proper to recognize

A charitable use, where neither law nor public policy forbids ...

In Bob Jones the Supreme Court found that racial discrimination in education is against public policy, but it did not clearly specify what made it illegal. The law that prohibits private schools from discriminating on the basis of race is not the Public Accommodations Act of 1964, but rather a provision of the Civil Rights Act of 1866, now codified at 42 U.S.C. Section 1981. This Reconstruction Era civil rights law prohibits racial discrimination in "contracting":

all persons ... shall have the same right ... to make and enforce contracts ... as is enjoyed by white citizens.

This statute helped to eradicate slavery. Under the institution of slavery whites owned African-Americans. The law of property did not apply to the slaves themselves. Slaves had no rights under the law of property. Nor did the law of contract apply to slaves. Slaves could not enter into or enforce contracts. Everything that the slave had or made belonged to the master, and only the master could contract for the labor of a slave. The Civil Rights Act of 1866 cured these terrible defects in the law. It was adopted by the same Congress that wrote the 14th Amendment. In fact, the Equal Protection Clause of the 14th Amendment was written primarily to make the Civil Rights Act of 1866 constitutional. No longer would the states be permitted withhold the protection of the law -- the EQUAL protection of the law -- to black citizens. Section 1981 of the the 1866 Civil Rights Act extends the protection of the law of contract to the newly-freed slaves, while Section 1982 extends the protection of the law of property to them.

For more than a century the Supreme Court ignored, eviscerated, and invalidated the Civil Rights Act of 1866 as well as other Reconstruction civil rights laws, but in Jones v. Alfred H. Mayer Co. (1968) the Supreme Court revived Section 1982 of the 1866 law, and eight years later in the case of Runyon v. McCrary (1976) the Supreme Court ruled that a private school that refused to admit black children had violated Section 1981's prohibition on the right to "make and enforce contracts" on a racially non-discriminatory basis. In Runyon the Supreme Court left open the question whether the law could be applied to a private school that practiced racial discrimination on account of religious belief. But in Bob Jones the Supreme Court closed that door, at least with respect to the denial of tax exemptions:

The governmental interest at stake here is compelling. ... The government has a fundamental, overriding interest in eradicating racial discrimination in education - discrimination that prevailed, with official approval, for the first 165 years of this Nation's constitutional history. That governmental interest substantially outweighs whatever burden denial of tax benefits places on petitioners' exercise of their religious beliefs. The interests asserted by petitioners cannot be accommodated with that compelling governmental interest ... and no "less restrictive means" ... are available to achieve the government's interest.

Could Congress, at some time in the future, outlaw discrimination on the basis of sexual orientation in private schools and universities? Yes, and I hope it does. Would the I.R.S. then be justified in denying tax-exempt status to schools that discriminate on the basis of sexual orientation? Yes, it would. Would those laws violate the constitutional and statutory rights of religious schools under the First Amendment and the Religious Freedom Restoration Act? That would depend upon whether the government's interest in either prohibiting acts of private discrimination against gays and lesbians or declining to subsidize such acts outweighs the burden placed that would be placed upon an educational institution that loses its unqualified right to practice such discrimination. (Note that non-discrimination laws do not apply to churches themselves, not would they apply on behalf of persons performing "ministerial functions" at religious institutions such as schools or colleges. Churches discriminate on all kinds of grounds, including religion, gender, sexual orientation, ethnicity, disability, and race, and they still receive preferential tax treatment.)

But Congress has not yet acted to prohibit acts of private discrimination by schools and universities or any other private parties based on sexual orientation. Until it does, the courts are unlikely to hold that the I.R.S. has the power to deny tax exemptions to private schools and universities that engage in that conduct, however deplorable.