Thanks for the suggestion, Igor.
I tried 5 IPs on the MaxMind site. One of them had a postal code record--the rest were blank. When using that IP with nginx/GeoIP, the postal code field was empty.
Using the MaxMind website, the city and lat/long of the IP are reported differently from the same fields from the free MaxMind database used by the GeoIP module. The MaxMind online lookup site states that the data is from their commercial product, as opposed to the free one that I'm using with nginx.
So...it seems that the databases are somewhat different, and there may simply be no postal code data in the free database for the IPs that I'm testing with.
Cheers