In going through my mail (from the gamedunio3 developer looking at doing HDMI output), and following some google links, I noticed there is an Arduino -> HDMI board out there, that has an optional Teensy 3.2/3.5 shield. I believe it may have 5v input to the Teensy, so the Teensy 3.6/4.0/LC may not be appropriate. I know nothing of the board other than the link:

I have to imagine that an alternate way to get HDMI output from the Teensy, would be to run a Raspberry Pi to do the HDMI support through its driver, and take serial, i2c, or spi commands from the Teensy. Of course it is just a simple matter of programming.