Do private religious institutions ever consider hiring someone clearly not affiliated with said religion for a faculty position?
You should check the particular university in question.  They will usually make it pretty clear in their job postings and/or application process.

Some will say explicitly "We welcome applications from all candidates regardless of faith", or "We do not discriminate on the basis of sex, race, ..., religion, ...".   The Jesuit institutions with which I'm familiar are of this kind.  

If they say that, it's likely that they mean it.  They'd have no particular desire to get applications from candidates that they don't intend to consider; it would just be more work for them.  If they did want to discriminate on the basis of religion, they wouldn't have to hide it, since it's perfectly legal under US law for religious institutions to do so.

You might still be asked to demonstrate that you share their values, even if you don't share their religion, and that you will be comfortable working in that environment.  You might look up the university's mission statement (they pretty much all have one) and see if it's something you can get behind.

Conversely, universities that only accept applicants of a particular religion will typically signal this in the job posting.  (Again, they want to avoid wasting your time or theirs.)  A common way is to require a "statement of faith" as part of the application, in which the applicant is expected to demonstrate adherence to the appropriate religion.  I recall another posting that asked the applicant to state their religion, "for instance, Methodist, Baptist, Presbyterian, ..." - their list only included denominations of Christianity, making it pretty clear that non-Christians need not apply.

You might also find this explained on a generic "about the university" page on their website.