Perhaps dumb, but would save a lot of hassle: are you sure the license is still valid? So not a temporary one that expired :).
Is the error in the Event log? It should tell you what URL it thinks it doesn't have the valid license for.
If you have the Always on setting in your azure app on, Azure does a periodic call to keep your process running. This is to the main url configured for the application. Often something like 'yourapp.azurewebsites.net'.. It could be that your reach a web farm limit because that one is also being called frequently. Either change the main url in the web app or setup a redirect for that azure url.. (or get a license for it :))
In Azure portal go to your Web App
- Select Configuration
- Select General settings
- Check the Always On setting