Does Exchange Server 2016 or 2013 allow the use of the Windows Server 2016 cluster CLOUD WITNESS feature?
https://docs.microsoft.com/en-us/windows-server/failover-clustering/deploy-cloud-witness
No, neither version supports the cloud witness option.
However, Exchange Server does fully support an Exchange File Share Witness server in Azure. If you choose this option, you’ll need a writable Domain Controller and a Windows Server to host up the File Share Witness directory. The reason for a Windows Server for the FSW, is that the ‘Exchange Trusted Sub-system’ group has to be in the local administrator group on the server. It is for this security reason that we do not recommend using a DC to host up an Exchange Server FSW. If you added the Exchange group to the local administrator group on a DC, that would set the group as a Domain Administrator Group, which should be as limited in membership as possible.