Only for specific, approved functions. Not because there's a religious angle.
So, for example, if Congress approves a deduction for donations to refugee operations, religious and secular organizations that donate to refugee aid should equally qualify for the approved operations. But not for any funds that might be skimmed off for non-refugee operations, be they salaries or proselytizing.
My memory of the details is faint, but the excellent Jane Meyer book
Dark Money looks at a number of ways that so-called charities and non-profits are really just scams to avoid taxes and/or to funnel money into political or self-serving agendas.