How to make flask server running in a VM externally available?
The problem isn't related to Flask, since you opened up your application to listen on any public IP (0.0.0.0).
Moreover you should do a proper port mapping in your azure configuration. Google said, you might have a look here: https://azure.microsoft.com/en-us/documentation/articles/virtual-machines-set-up-endpoints/
EDIT
Another idea, where some colleages often run into, is that you may have skype open which somewhat uses port 80/443 and therefore is blocking it. Shutdown skype if you do so or use a different port for your webapp.
So here's the deal with Azure:If you open a port from the Azure portal, the firewall in your VM STILL blocks that port. You have to manually go in and create a firewall setting in your VM to keep the port 5000 open for your flask server. Once that is done, you should be able to connect to it.