Your problem is related to HDCP and EDID. When you turn off a monitor or projector (HDCP sources), the Denon must renegotiate copyright protection between just the Firestick and the Denon with no display in the mix.
When this change occurs, the Firestick automatically pauses playback.
Similarly, when I unplug headphones from my iPad and a show is playing, it automatically pauses playback, instead of continuing playback through the speakers. I must press play again.
If you have a remote control app for your Firestick, you likely can press 'play' on it after five or ten seconds of the projector being powered off.
This is NOT a failure on the part of the Firestick, or your Denon. It is just meant to be a feature, and likely isn't one you can turn off (go team stupid programmer!). So, you will need to find a workaround, and that is most likely by using the remote. Not sure what remote it comes with, but you may also just try to press the 'play' button to see if that starts audio back up. But, I would think the app for the Firestick may provide additional details of operation.
I haven't looked on Google for solutions to retain playback on display shutdown, but I wouldn't be surprised if this question hasn't been asked elsewhere already.
It won't help to disconnect or reconnect anything to your receiver. That's definitely not the issue IME.
EDIT: Did a quick Google, and here is something that confirms what you said, and indicates the fix as simply pressing play again on the remote...
https://www.reddit.com/r/fireTV/comments/4adg35