Diagnosis
This is almost always a handshake issue between the projector and your source device. They're not agreeing on a video signal format when they first connect.
How to Fix It
Let's reset the conversation. 1) Turn the projector OFF with the power button. 2) Unplug the HDMI cable from both the Fire Stick and the projector. 3) Unplug the Fire Stick from its power source. 4) Wait 30 seconds. Now, plug everything back in, in this order: First, plug the HDMI cable firmly into the projector's port. Then plug it into the Fire Stick. Next, power the Fire Stick. Finally, turn the projector ON. It should go through its search cycle and find the signal. If it doesn't, press the 'Source' button on the remote until you select the correct HDMI port you're using (HDMI 1 or 2).
If That Didn't Work
The issue could be the HDMI cable itself or a setting on the projector. Try a different, known-good HDMI cable—preferably a 'High Speed' one. If you have another device, like a game console, test that to rule out the Fire Stick. If another device works, the problem is with the Fire Stick. Go into the projector's menu: Settings > General > HDMI Ultra HD Deep Color. Try turning this setting OFF for the port you're using. Sometimes turning this advanced feature off fixes handshake problems with older streaming devices.
Heads Up
If you're using a very long HDMI cable (over 15 feet), it might not support the 4K signal reliably. For long runs, you might need an active or fiber optic HDMI cable.