Azure App Service (https://azure.microsoft.com/en-us/services/app-service/) is Azure's Heroku competitor. It's quite close in features and ease-of-use. Disclosure: I work for this team.
Also, one of the killer features of Heroku (which few services seem to replicate) is log drains - I can easily add a http or syslog endpoint and have Heroku send the logs over. The other killer feature which isn't often replicated is One-off dynos, where we can spin up a new instance and get a console attached to it in one command - useful for running database migrations or using Ruby as a CLI to access data.
If we were on .NET that would probably be attractive, but it's still not really competing with Heroku.