Diagnosis
This is an EDID handshake issue. The projector's HDMI ports are configured by default for standard HDR10. The connected device is sending a Dolby Vision signal the projector is not prepared to accept on that input.
How to Fix It
You must manually enable Dolby Vision compatibility for the specific HDMI port. Go to Settings > Display & Sound > HDMI. Select the HDMI port your Apple TV is using (e.g., HDMI 1). Change the 'HDMI Format' from 'Standard' to 'Enhanced' or 'Dolby Vision'. The Apple TV video settings should then be set to 4K SDR 60Hz with 'Match Content' turned on for both Range and Frame Rate.
If That Didn't Work
The HDMI cable is likely the bottleneck. You require a certified Ultra High Speed HDMI cable (48Gbps). Older 'High Speed' cables (18Gbps) cannot carry the full Dolby Vision signal.
Heads Up
Enabling Dolby Vision mode may cause compatibility issues with other non-DV devices on the same port. You may need to switch the HDMI format back when changing sources.