The new Raspberry Pi Zero doesn't have any kind of wired Ethernet connection. Has anyone have tested if Ethernet over HDMI works? I did not find any info in the web or even what version of HDMI the Raspberry Pi Zero uses.
No, the Pi Zero does not support ethernet over HDMI.
Unfortunately the schematics for the Pi Zero are not available (edited: an overview schematics is available now, see updated information below). However since it is essentially an minimized Pi 1 I'd bet that the HDMI circuitry is (nearly) the same - though the Pi 1 features a standard sized HDMI connector (Type A) whereas the Pi Zero comes with a mini connector (Type C).
Looking at the schematics of Model A and B (Type 1) Revision 2.1 on Github or on webarchive (Raspberrypi.org's schematics are now downgraded to less detail), page 2, we find that the pin 14 at the HDMI connector is not connected. This pin is responsible for the HDMI-Ethernet-Channel (HEC) for HDMI 1.4 and reserved for HDMI 1.0-1.3. So there's no HDMI ethernet connected on the Pi.
Pin 14 (reserved for HEC) on the Type A connector corresponds to pin 17 on a Type C connector, see Wikipedia. The overview schematics of the Pi Zero 1.3 shows this pin as not connected (thanks to user Wilf for pointing to this updated information).
On a side note: Two facts support the notion that there is no HEC on the Pi Zero - there is no dedicated ethernet controller on the board (such as the LAN9512 for the B/B+/2B Pi) and to be honest, if the Pi Zero would support HEC, the Foundation would have announced it as such! It's just not a feature that you hide.