OK, after some more testing (which included turning off Energy Efficient Ethernet) I believe what was causing the problem was changing which HDMI port the monitor was plugged into.
The unit exhibits the problem when the HDMI port adjacent to the ethernet jack is used. If I use the HDMI port fathest away from the ethernet jack it works fine.
I'm very happy with my setup of Intel DC3217IYE NUC & Akasa Newton fanless case. After putting the NUC into "Low Power Mode", venting the case's cover, & using the case in vertical orientation the idle temperature is around 48C and peaks 65C under full load (which the unit never really is under, most of it's tasks only put it under a 10% load).