Keep it on, especially if you have a Mi Box ... you never would want to miss an update for it.
Keep it on, but make sure it goes into sleep mode when youo turn your TV off. "On" is different from "Woken up and in full power mode." It'll wake up in a couple of seconds, but booting from power off can take up to 5 minutes.
They have fans you can buy for boxes. Bought few for myself. Have nivida shield and mibox. Both never get as hot even with heavy usage.
I have kept my shield on 24/7, except for occasional reboot because there seems to be something wrong with the way my new Philips tv uses hdmi-cec commands.
Sometimes it starts to work only when I restart it through the power menu.
No is bad
These boxes are meant to remain always on for easy access. Most of them have an idle state that uses very little power. The Shield can even be used as a Plex server, so needs to be on.
Of course if you live in a hot country with no a/c, you need to make sure to improve the box's heat dissipation