I have two setups in my house. On both, I use a Thunderbolt 2 compatible adapter that connects to the firewire port and gives me an HDMI output signal. Then I run an HDMI cable from there to an HDMI switch by Kinivo. The output runs from there to an HDMI 1080p HDTV.
One uses a MacMini from late 2009 running El Capitan (10.11.6), which seems to be the latest I can upgrade it to. That particular setup uses an Epson 1450 Home Cinema projector. I've tested it with other HDMI TVs, though. When I switch the HDMI switch to Output 1 (the MacMini), I see the MacMini display with no issue.
On the 2nd setup, I have a 27" iMac, from late 2013, running macOS Sierra 10.12.6. (Haven't had time to update to the latest yet.) It's the same setup, but, of course, to a different 1080p HDTV. If I run the HDMI cable from the Thunderbolt 2 display adapter to the TV (and I've tested it with different TVs), I get a clear image and can use the TV as a 2nd monitor. But once I run the signal from the adapter, through one cable, into the HDMI switch (same brand, same model, as the other one - ordered them together), on the HDTVs I've tested it on, I get "Unsupported Signal."
The factor that changes is adding the HDMI switch.
I also have an input on both HDMI switches from a DISH TV device. In all cases, on both systems, when I switch to the DISH device as an input, everything is fine. Inputs from blue ray players are fine. But on this 2nd system, with the iMac, whenever the signal goes through the HDMI switch, the result is "Unsupported Signal."
Can someone give me an explanation of what is happening and if it's fixable. What puzzles me is this happens on one switch and not the other. (I'm hoping to swap out switches at some point to test them.) I'm wondering if the output on the iMac could be different than the output from the MacMini.