Of course you can send data out of a Pi at huge data rates using HDMI. That is what it is for, getting data to your monitor.
At it's simplest all you have to do at the Pi end is write data into the screen buffer device. And boom it ends up on the screen!
$ sudo cat /dev/urandom /dev/fb0
(Or whatever the frame buffer number is)
You should see a lot of nice random "snow" on the screen.
Now, actually connecting that HDMI to something other than a screen and making use of it is a bit more tricky.
One approach is to use an FPGA to capture the HDMI. The you can do whatever you like with the data stream.
For example: https://github.com/timvideos/HDMI2USB
All it needs is to buy an FPGA board to use it on. Like so: https://store.digilentinc.com/atlys-spa ... exys-video