For argument's sake, I tried running it from the command line but starting in the app directory, to make sure it wasn't some weird path issue. IOW:
% cd /Applications/OpenLP.app/Contents/MacOS/
% ./OpenLP
That works just fine too, no difference. Still can't open via double-click. So, for a workaround, I created a file named "OpenLP.command" in my home directory, with one line in it:
/Applications/OpenLP.app/Contents/MacOS/OpenLP
I can double-click that file, it auto-launches in Terminal, and OpenLP runs without an issue.