We’ll, depending on projector, it’s use may be somewhat stunted, but it should work fine
When it comes to video, no problems whatsoever, it will work on a projector the same way as it would work on a TV, just plug it into power and one of the projector’s HDMI ports, switch to that HDMI port and you will see the Fire TV stick display on the wall/screen
Now here’s why I said it may be somewhat stunted depending on projector, when it comes to audio, you may run into some issues
There is no secondary audio out on a Fire TV stick, it’s HDMI only, the problem? Not all projectors have speakers (or audio out for that matter), so depending on projector, there may not be a way to get any audio from the Fire TV stick at all
The thought process behind this is the manufacturers of home theater grade projectors (as in not projectors designed for office use, like for presentations) don’t see their customers connecting their devices directly to the projector, most of their customers are going to connect their devices to an AVR, and run one HDMI cable from their AVR to their projector, so since the AVR takes care of the audio, why bother with speakers and/or audio out?
This means if the projector doesn’t support audio, you will need to find a device to extract the HDMI audio before connecting to the projector