I just upgraded 2 of my PC's to Office 365 Business, which includes Outlook 2016, from Office 2010. I'm doing the local install and have and Exchange 2013 Server in house. I'm testing my own stuff before upgrading other PC's in the company.
With outlook 2010 and Exchange 2013 Server (on premises) I had several high-volume users using the "Online Archive" option for their older emails. This includes myself.
After the update, my current files and profile seems to load fine, just the online archive is missing. I also did another PC to verify this issue. I don't see any options to enable it. My other laptop which is not upgraded from Outlook 2010 still shows the archive as does my OWA login. Several searches have turned up nothing.