How does copyright apply to a small church?

It applies to them pretty much the same way it applies to anyone else. In my experience, publishers of church music guard their copyrights every bit as fiercely as any secular publisher.

US Copyright Office on Fair Use

Fair use has four parts (see link), one of which is effect on the publisher’s bottom line. If churches are allowed to freely photocopy whatever they want for their own use instead of buying hymnals, etc., that would seriously affect publishers’ profits.

Older songs may be in the public domain.

Using copyrighted material without paying for it is theft.

As noted above their size or type of organization does not allow them to break copyright laws.

Agree with the above. In addition, however, when a prospective user of copyrighted material writes for reprint permissions, it’s ok to mention intended use and audience and ask for a waiver of fees or negotiate a fee. They may not be granted—I remember once asking on behalf of a newsletter with a circulation of 50 and being told that the fee (several hundred dollars) was nonnegotiable; and so we didn’t use the item.

It might be worth considering whether there is already in existence a songbook that might contain enough of the songs the church is interested in using to be a good investment for them.

My old Baptist church had bibles that were copyright to a maximum of 250 words per sermon plus they had to pay a fee to show the lord of the rings clip.

@Hawaii_Jake “Using copyrighted material without paying for it is theft.”
No. Using copyrighted material is just using it. Copying it may not be allowed, unless you get permission. The church might try asking permission, or using other songs.

@Zissou “If churches are allowed to freely photocopy whatever they want for their own use instead of buying hymnals, etc., that would seriously affect publishers’ profits.”
That’s a common conceit that industries use to try to exaggerate their “losses” in order to try to justify copyrights. But such a claim would only be true if, faced with the alternative of paying the copyright holder, the people would choose to pay the copyright holder. If instead they find something else to sing, the effect is just to have people sing different songs.

@Zaku If your point is that interpretation of fair use has swung too far in favor of copyright holders, I agree. For that matter, if you want to go farther and say that the fundamentals of intellectual property law need to be reexamined in the light of current technological and economic conditions, I agree. But the OP is asking what to expect under current law.

@Zissou My point is both what you just wrote, and that the existing answers have also overstated both the reasoning and the practical parts of their answers due to what I feel is excessive acceptance of the thinking generated by the industry trying to influence lawmakers and people into thinking of not just copying but even use without paying as theft (as seen in Hawaii_Jake’s answer) and that using something without paying a publisher is always impacting publishers’ profits (your answer). Also, as a church in particular says it is a moral institution, I’d say it’s important to be clear on such points.