If we're talking reserve funds, then two factors need to be examined that have an inverse relationship one with the other: access and rates. Let's say that your church has enough reserve funds to cover a year's worth of expenses, but you only really need immediate access to 3 months worth. With the 3 months of reserves, it really doesn't matter whether you choose a bank or credit union because you'll want quick, easy access to those funds. So, you'll put the funds into a savings account or a money market account at best. For the remaining funds, you may consider something like CD's, which offer a higher rate, but often put (limited) restrictions on access- meaning that there may be an interest penalty or surrender charge or something like that.
If your church has sufficient reserves to do so, consider a tiered and layerd technique. The first tier of your reserves should be money you can get the easiest access to, to handle emergencies. The second tier has similar access, but higher rates (Money market account, certificates with no penalties, etc.) With the third tier, you can consider a layered certificate approach. In other words, purchase certificates that have different maturities- 3 month, 6-month, 1-year, etc. With this layering strategy, you can gain access to higher rates of return without sacrificing access to the bulk of your funds. An example is to have a CD mature every 3 months, so let's say you purchase:
1- 3 month CD at 1.5% (hypothetical)
1- 6 month CD at 1.75%
1- 9 month CD at 2.1%
1- 1 yr CD at 2.7%
When the 3 month CD matures, you roll that into a 1 yr CD. Why? Because you have another one coming due in 3 months at a lower rate. With each subsequent maturity, you purchase a 1 yr CD. Eventually, you'll always have money coming due every 3 months, but now at the 1 yr rate, as opposed to a 3 month rate. Even if you had to withdraw all of the funds for a huge emergency, the penalties are normally very low, and normally don't impact the principle at all. (Usually, there's an interest penalty) So, you lose nothing (except for the opportunity cost).
Sorry for the long explaination. To me, it doesn't really matter what type of institution you use. All of the funds are going to be FDIC insured up to certain limits anyway. With all of the mergers and acquisitions that are going on, who you bank with today may not be who you bank with tomorrow anyway.