static void DeleteFolder(ExchangeServiceBinding esb)
{
    // Create the request.
    DeleteFolderType deleteFolderRequest = new DeleteFolderType();
    deleteFolderRequest.DeleteType = DisposalType.HardDelete;

    FolderIdType[] folderIDArray = new FolderIdType[1];
    folderIDArray[0] = new FolderIdType();
    folderIDArray[0].Id = "AQApAH";
    deleteFolderRequest.FolderIds = folderIDArray;

    try
    {
        // Send the request and get the response.
        DeleteFolderResponseType deleteResponse = esb.DeleteFolder(deleteFolderRequest);

        //// Get the response messages
        ResponseMessageType[] rmta = deleteResponse.ResponseMessages.Items;

        foreach (ResponseMessageType rmt in rmta)
        {
            // Cast to the correct response message type.
            if (rmt.ResponseClass == ResponseClassType.Success)
                Console.WriteLine("Folder deleted from mailbox.");
        }
    }
    catch (Exception e)
    {
        Console.WriteLine("Exception: " + e.Message);
        Console.ReadLine();
    }
}

<DeleteFolder xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" 
                  xmlns:xsd="http://www.w3.org/2001/XMLSchema" 
                  DeleteType="HardDelete">
  <FolderIds xmlns="http://schemas.microsoft.com/exchange/services/2006/messages">
    <FolderId Id="AQApAH"  xmlns="http://schemas.microsoft.com/exchange/services/2006/types" />
  </FolderIds>
</DeleteFolder>

<DeleteFolderResponse xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" 
                      xmlns:xsd="http://www.w3.org/2001/XMLSchema">
  <ResponseMessages xmlns="http://schemas.microsoft.com/exchange/services/2006/messages">
    <DeleteFolderResponseMessage ResponseClass="Success">
      <ResponseCode>NoError</ResponseCode>
    </DeleteFolderResponseMessage>
  </ResponseMessages>
</DeleteFolderResponse>