The customer in this case had an Archive policy set on the mailbox that should move all items older than 1 day to archive, and items were not moving at all.
We went through a lot of "typical" troubleshooting that we would look at when items are not being moved to the archive. (An example of this would be the information in https://blogs.technet.microsoft.com/ehlro/2014/01/15/more-on-exchange-retention-and-archiving-policiestroubleshooting/)
After consultation with one of our Escalation Engineers that works with the Exchange Online engineering team responsible for the MRM/Archive features, I was able to get the information I needed to help the customer solve the problem.
Normally when troubleshooting whether or not items have a retention/archive policy assigned, we will look at the properties of individual items with MFCMAPI to see what expiration information they have on them. I found out in this case that when we have a Default Move to Archive tag applied, we do not actually tag the items, but we create a Search Folder in the mailbox to "query" for the items that should be moved.
The specific instructions for checking the Search Folder were:
Use MFC Mapi and go to Root Container -> Finder and open the folder NoArchiveTagSearchFolder...
When you open the folder it should show all items that do not have an archive tag - these are the items that MRM would process based on the default tag.
Sort by Receive date. Are there any items that should expire? If the folder does not exist, then MRM may not be recognizing the default archive tag.
The folder did exist here, but there were no items showing up in it, even though there were plenty of items.
We deleted the search folder with MFCMAPI, and then ran the following command in PowerShell connected to Exchange Online (https://technet.microsoft.com/en-us/library/jj984289(v=exchg.160).aspx):
After this, items started moving to the archive as expected. We also could check with MFCMAPI and see the search folder now showed items that should move to archive, but had not yet moved.