No one, not even the teacher, can delete a Post Office message. The reason for this seemingly odd design decision is similar to the reason why most messages in the Assignments module can't be deleted (see What can the teacher delete?). If Post Office messages could be deleted, the following nonsense would occur:
Someone will claim they sent you an important Post Office message, which for some strange reason you never received.
Someone will claim they never received your important Post Office message.
Someone will claim they might have received your important Post Office message, but they must have accidentally deleted it before reading it.
...and so on. All of the above claims are routinely made with ordinary Internet email. Because Post Office messages can't be deleted none of the above are possible. In fact, in the Post Office:
Messages are always received by the recipients with 100% reliability. If you can read the message in your Outbox, the recipients can read the message in their Inbox. Guaranteed.
No one can accidentally (or deliberately) delete a message. Once a message has been sent, it's a matter of record.
It's interesting how the reliability of technology seems to suddenly improve when you change software to force people to be honest!
While you can't delete a Post Office message, you can store it in the "Attic". To permanently store a message in the attic, click on the Store in Attic button while you are reading the message:

You can work with messages stored in the attics as if they were any other message. Recall there are two attics, one for the Inbox and one for the Outbox. To enter an attic, click on the Go to Attic button:

Finally, note that the messages stored in the attics are included in the pages displayed by the Post Office "Gather" commands. (See Gathering Messages for a general description of how to gather messages.)