I know in-place updates of Azure VM's are not supported by Azure based on https://stackoverflow.com/a/41924158/2713516, but I'm still wondering if I can run an in-place update myself by simply downloading the Windows Server 2016 installer and running it on my Azure Virtual Machine that currently runs Windows Server 2012 R2.
Will this break any Azure mechanism's I'm not aware of?
Based on my knowledge, Azure does not support this.
All Microsoft, software that's installed in the Azure virtual machine environment must be licensed correctly. By default, Azure virtual machines include a license for using Windows Server in the Microsoft Azure environment. Certain Azure virtual machine offerings may also include additional Microsoft software on a per-hour or evaluation basis.
Even you upgrade successful, you need a Windows server 2016 license. Azure does not know you have upgraded your OS, original license will be failure.
More information about Virtual Machines Licensing FAQ please refer to this link
Also, you could check Converting a current retail version to a current volume-licensed version
