Solution: Stop using the "Default (Azure provided)" DNS service
This worked for us so I am posting it as an answer.
- From the Azure portal, select your Azure Function, and go to the Networking page (that's on the left side menu now, but they "improve" the interface all the time so it might be somewhere else)
- Under "Outbound traffic configuration" go to "Outbound DNS"
- Change from "Default (Azure provided)" to "Custom" and add the DNS server addresess of your choice. We used the well-known Google servers at 8.8.8.8 and 4.4.4.4.
- Don't forget to save this setting, and you will likely have to restart your Azure Function
This worked for us immediately, and the usually-intermittent (but today, consistent) DNS errors disappeared. For details on our rig, please see my comment on the original question.