Wikipedia says the Raspberry Pi supports "HDMI (rev 1.3 & 1.4)" and the HDMI 1.4 specification has Ethernet. Could you have a HDMI 1.4 to Ethernet adapter? and would the transfer rate be faster than USB 2.0?
As stated and tested here, Raspberry PI does not support HDMI Ethernet Channel.
Looking at the schematics of Model A and B Revision 2.1, page 2, as well as the schematics of Model B+ Revision 1.2, page 1, 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 A/A+/B/B+ (and presumably neither on the Pi 2 and the Pi Zero).