While I have some probably unrelated HDMI issues (sporadic resync only if HDMI is connected via AVR receiver in standby passthrough mode), I don't have any network issues independent of the HDMI port in use.
I just did some synthetic network tests (NetIO) and copied an 8GB file to the NUC. No matter which HDMI I use to display the desktop (@1920x1080, 59p), the network performance was identical and very close to the theoretical limit (~113MB/s).
So from my point, there is not a general issue with Ethernet and HDMI even if the ports next to each other are used. Cable shielding could be an issue, especially if the cables run in parallel.