To export a Exchange 2010 mailbox this can be done through the Exchange Management Shell.

NOTE: This runs on Exchange Server 2010 SP2 and later.

New-ManagementRoleAssignment -Role "Mailbox Import Export" -User Domain\User

or if you want to do it via Active Directory group based. You can run:

New-ManagementRoleAssignment -Role "Mailbox Import Export" - Group "Group Name"

After this you might have to close out of the Exchange Management Shell and open it back up for the newly created permissions to take effect.

After that you need to have a destination. We’ve had success with just exporting the PST locally to the Microsoft Exchange Server’s local drives.

1. Create Folder on a local drive – C:\PST (or where ever you like)
2. Add Exchange Trusted Subsystem Group to the NTFS permissions and select the Modify Permissions Level.
3. Share out the folder and provide Write Access. (We normally just do it on the everyone group because the share isn’t valid for long.)

After this run the cmdlet (the line below is if you are running it on the Exchange Server that you created the share (destination location) on:

New-MailboxExportRequest -mailbox <users-alias> -filepath \\<localserver_name>\pst\user.pst