@ BeenThere - one factor to consider is that editors ultimately make decisions on manuscripts and they may be able to see the author(s) name(s) and affiliation(s). In general, an editor has some flexibility whether an article winds up with a "reject" or "revise and resubmit". That said, I'd like to hope that there is some lack of bias in academia when it comes to affiliation - but as a recent Perspectives in Psych Science article noted about which factors help grad students in landing tenure-track positions, apparently affiliation is a factor.
@ The original question - I'm at a smaller liberal arts college that's not particularly well-known outside of my geographic region. It, too, has some religious ties but is not a "religious school." So far, I've not noticed a major impact in decisions regarding my manuscripts or conference presentations. And, so far all of my submissions have been accepted at major conferences (including submissions without any of my former grad school colleagues' "big" names attached).
The good news, if there is any, is that with the poster submission being rejected - you and your students can resubmit elsewhere. See what other regional (or heck, even national) conferences have approaching submission deadlines and send it in somewhere else. You might be able to take it to an even better venue. The most important thing when it comes to publishing or submitting presentation abstracts is to never give-up unless you have good evidence that your work has a serious flaw. Odds are the submission will eventually stick somewhere!