In the application log there are errors from ServiceDesk.MailOutbound that say "Object reference not set to an instance of an object."
In the Outbound Mail Server Settings you can check the box for "Outbound emails to use Inbound Mail box". Once this is checked when you send an assignment or reminder from the process it checks to see if there is a mapping associated with the process. If there is then it checks the mailbox for that mapping and sends the mail message from that mailbox instead of sending it from the default mailbox. If there is a mapping but no mailbox that uses that mapping then it will not be able to detect what mailbox to send the email from and you will see these errors in the logs.
Deleting the mapping for that process will fix the problem. Once it does not detect the mapping it will send out a message from the default mailbox without any furthere errors.