Diagnosis
The projector isn't detecting the source because the Fire TV stick's output resolution or refresh rate is incompatible with the projector's current power state or input setting.
How to Fix It
With everything plugged in, press the 'Source' button on the projector remote until you cycle back to the correct HDMI input. If that fails, power cycle the projector with the source already on and plugged in.
If That Didn't Work
Plug the Fire Stick into a TV, lower its output resolution to 1080p 60Hz in its display settings, then reconnect it to the projector.
Heads Up
Some streaming devices don't output a signal until you're fully logged into their home screen. Ensure the device is fully awake and active.