Within the last week or so, my Lightburn asks me if I want to update to the most recent version. I accepted and downloaded the file.
When I go through the install from the file it will open the newest version and the message goes away. However if I close the program and relaunch it from the tray, it asks me again if I want to update to the newest version.
What am I doing wrong?
You may have both the older and the newer versions still on your Mac. If you look in the Applications folder, do you see more than one instance or listing for LightBurn? You may be launching the older one from your tray, and that would trigger the newer version available message again. 
I will have to check that. I’ve been looking at the “Launch Pad” and it only shows one version. Maybe in the Apps folder there are two.