The policy, contained in a Justice Department memo from 2007, said that under the Religious Freedom Restoration Act, religious groups seeking federal grants were exempted from complying with non-discrimination laws in hiring.

BuzzFeed:

The 130 organizations — including the ACLU, Anti-Defamation League, Human Rights Campaign, NAACP, and Planned Parenthood — counter in their letter to Obama that the opinion “relies on flawed legal analysis” that RFRA “provides a blanket override of a statutory non-discrimination provision.” A copy of the letter was provided to BuzzFeed News. Members of the group have been asking the Justice Department or White House to rescind the OLC Memo since Obama took office, but this is the first time that the letter has been sent directly to the president. Previous letters were sent to then-Attorney General Eric Holder, as in 2009 and 2014, as well as to the head of the White House Office of Faith-based and Neighborhood Partnerships in 2013. In the new letter, the group tells Obama: RFRA was intended to provide protection for free exercise rights, applying strict scrutiny, on a case-by-case basis, to federal laws that substantially burden religious exercise. RFRA was not intended to create blanket exemptions to laws that protect against discrimination. Yet, in contrast to this, the OLC Memo relies on flawed legal analysis and wrongly asserts that RFRA is “reasonably construed to require” a federal agency to categorically exempt a religiously affiliated organization from a grant program’s explicit statutory non-discrimination provision, thus permitting the grantee to discriminate in hiring with taxpayer funds without regard to the government’s compelling interest in prohibiting such discrimination.

David French of NRO explains what's at stake:

In plain English, the letter — signed by the usual alphabet soup of leftist special interests – is asserting that the federal government has a “compelling interest” in prohibiting a religious organization from using religious criteria when hiring its employees, including key leaders of the organization. The Bush-era letter stood for the simple proposition that a religious grantee could only those people who affirm and practice the group’s faith. Unless one is consumed with malice for orthodox faith, this is one of the most common-sense principles imaginable. Precisely what real state interest — much less compelling interest – is preserved by forcing Jewish groups to hire Muslims, Christian groups to hire Hindus, or requiring Christian groups to retain employees who flout Christian doctrine and practice? Religious charitable organizations tend to be the most effective organizations in the country in helping the poorest and most vulnerable citizens — and in helping those who struggle with drug addiction or are attempting to build new lives out of the ruin of a criminal past. They are effective precisely because of their faith, not in spite of it, and if the government undermines a group’s faith, it will also undermine its faith-based work. A government seeking to intelligently spend money on private grantees would do well to spend money where it will do the most good. But none of that matters to the Left, because somewhere a Christian group — for faith-based reasons – might be asking its employees to dissent from any given doctrine of the sexual revolution. And we can’t have that, because sex is god.

Repealing the policy wouldn't necessarily "force" a Christian group to hire a Hindu or Muslim, but it would allow a non-Christian to sue the organization for discrimination. And since the burden of proof in such cases is on the defendants to prove they aren't discriminating, such a suit would have a decent chance of being successful.

Nothing would please the left more than seeing Christian groups fending off lawsuits from Muslims who will claim "discrimination" when they aren't hired. And it fits in nicely with their worldview, where all religion in America will eventually come under the control of government. They care about discrimination only as a weapon to use in this quest, punishing Christians because they have the gall to disagree with them.

Obama is just the president to help them bring this about. But whether he will endure the firestorm that would erupt if he repeals the policy remains to be seen.