Minimum Fee for Membership Types won't save
After spending time to resolve this across several days and working with our site administrator to troubleshoot we figured it out. The site administrator was able to confirm that the values were being stored in the database, so it was probably something to do with formatting.

I eventually discovered that the display rules were configured wrong. It was configured as follows:

Monetary Amount Display: %c %a

Monetary Value Display: %c

As I understand it, the formatting rules here are nested. When you're trying to display currency, it first looks to "Monetary Amount Display" to figure out how to display the value. In the case above, it was telling it to display the currency symbol with %c followed by the amount with %a.

The second field, "Monetary Value Display" contains the rules for displaying the value requested by %a. With the example above, the rules were to show the currency symbol for the value, which was invalid, so it chose not to show values at all.

I corrected the setting as follows:

Monetary Value Display: %!i

This instructs CiviCRM to display the value using the localization rules specified earlier in the form but to suppress the currency symbol. We don't need it, since we're adding a currency specific symbol in the "Monetary Amount Display" field.

I hope my frustration will save someone else some pain later on!