During a web app migration project our customer had questions about breaking changes introduced by Microsoft during maintenance of the Azure App Service infrastructure. The pleasant thing about Platform as a Service is, that you don't have to worry about the underlying operating system and its components. But Azure still runs on servers that must be patched so this is a valid concern.
A quick search on google produced a forum post by Cristhian Uribe that confirmed my initial assumption: Microsoft will not update the platform (i.e. .NET 4.6 or Java 8) you configured in the web application's settings (because that would be crazy, right?). If you set a Java app to use the newest Java Minor version or the newest Tomcat 8.5, it is your responsibility to check if your app is fine with that configuration. But wait, there's more! Under the hood, most of Azure's App Service plans run Windows + IIS and we all know that patching is inevitable here. Here's the quote from the forum post on this topic:
we do update the OS and IIS version from time to time and when we do if we see the possibility of this affecting your app there will be an e-mail sent to the e-mail account registered under the subscription notifying him/her of the maintenance. Normally you shouldn't experience any downtime.
Summary: Microsoft will not break your apps and if there is only a slight chance that this might happen they'll give you a heads up.