has been corrupted, and it's in an inconsistent state

Error Deleting Database "Failed to remove monitoring mailbox object"

When removing databases from Exchange 2013, you might get the following error if the correct procedures are not followed:

Failed to remove monitoring mailbox object of database “database_name”. Exception: Active directory operation failed on “server_name”. This error is not retrievable. Additional information: Access is denied. Active directory response: 000000005: SecErr: DSID-031520B2, problem 4003 (INSUFF_ACCESS_RIGHTS), data 0.

In this case, the database was removed an Active Directory [AD] error (with a not very useful description) complaining about insufficient permissions is thrown. If you run:

Get-Mailbox -Monitoring

You will most likely see a warning regarding a corrupted Health Mailbox:

WARNING: The object “domain_name”/Microsoft Exchange System Objects/Monitoring Mailboxes/”Health_Mailbox_GUID” has been corrupted, and it's in an inconsistent state. The following validation errors happened: WARNING: Database is mandatory or UserMailbox.

Because Exchange 2013 did not have sufficient permissions to the domainname/Microsoft Exchange System Objects/Monitoring Mailboxes Organizational Unit [OU], it could not delete the AD objects related to the database’s health mailboxes. In this case, the database attribute is null because the database the health mailbox references no longer exists.

To fix this issue, simply delete the health mailboxes referenced by the error(s) from that OU by using Active Directory Users and Computers. After removing these, the warning should be gone.

Deleting health mailboxes is a low risk procedure because they will be automatically re-created by the Microsoft Exchange Health Manager service on the Exchange 2013 server hosting the database when this service is restarted.