Old, i know but still relevant.
One important thing to consider is not to use mongodb. I don't know if Azure has a strange config that doesn't go well with nodebb's standard config or if it's just a bug in their current version (they use 3.2, current is 3.6 on Ubuntu, 4.2 on Windows).
It works with redis (at least connection from my VM succeeded).
Also julienstroheker managed to get an older version of nodebb running in an azure App Service. So basically there is a way 🙂