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So I know this thread is older, but I thought I'd chime in, as I travel with a FireTV and have had to resort to increasingly ridiculous workarounds to get hotel TVs to accept my illicit HDMI input.
I just came across a newer one and thought I'd share. I travel for business and have been enjoying the newer TownePlace Suites by Marriott chains for many reasons, but the TVs were actually kind of nice. They have an LG hijack box that has several streaming apps you can log into with your own credentials that are automatically cleared on checkout. Many of them also have a Chromecast attached so you can stream from your mobile device for apps that aren't represented on the built-in box. This has meant that my Fire stays in the suitcase, which is fine; one less thing.
However, the most recent one I stayed at was brand new, and the Chromecast hadn't been configured yet, and they were missing the HBO Go/Now app, boo! So I wanted to plug in my FireTV. Problem: the HDMI ports on the TV are disabled. You can pull the hotel's HDMI out of the TV and try to use that, BUT the LG-hijack box controls volume with the non-standard remote, and there are NO buttons in the TV itself to be able to change it (and it defaults loud enough for your neighbors to call the front desk on you).
Workaround: Look behind the TV, behind the dresser/stand it's bolted to, if you can. The LG-hijack box is bolted back there and IT has two HDMI inputs. One contains the (unconfigured) Chromecast, and one is empty. I wasn't able to make it work in the empty one, but replacing the Chromecast with the FireTV enabled me to operate as normal.
Just remember to put it all back as you found it before checking out; don't be that guy.