This is often seen when mailboxes are moved between servers or in a DAG/CAS array failover but this was not the case this time. The only changes made were deletion of the public folder database on Exchange 2007 and decommission of the Exchange 2007 server.
In the Exchange 2007-2010 days, you'd set the default public folder database for each mailbox database by using Set-MailboxDatabase so I checked to see what this setting was on Exchange 2013 and found that it referenced the deleted public folder database:
You cannot use either of the below commands to remove the setting:
Set-MailboxDatabase -PublicFolderDatabase $null
Set-MailboxDatabase -PublicFolderDatabase ""
To remove the setting from the mailbox databases, you need to use ADSIEdit. To do this, open up ADSIEdit and connect to the configuration naming context:
Browse to CN=Services > CN=Microsoft Exchange > CN=Organization Name > CN=Administrative Groups > CN=Exchange Administrative Group (FYDIBOHF23SPDLT) > CN=Databases.
For each database open properties and clear the value for the msExchHomePublicMDB attribute:
This should resolve the issue.