Don't think there's any reason for having those API calls in your code, unless you know it is there for a certain reason.

So here's what I think you are after.

Code:
Sub TicketResolved()

    Dim myOlApp As Object 'Outlook.Application
    Dim MyItem As Object 'Outlook.MailItem
    Dim Lastfirstname1 As String
    
    Lastfirstname1 = "SOMETHING"
    Set myOlApp = CreateObject("Outlook.Application")
    While Not IsEmpty(Cells(3, 3))
        Set MyItem = myOlApp.CreateItemFromTemplate("C:\OPEN -ALL\TICKET RESOLVED\Ticket Resolved Template")
        With MyItem
            .To = Cells(3, 2) & "; forums1167@gmail.com"
            .Subject = "Ticket Resolved #" & Cells(3, 1)
            .HTMLBody = Replace(.HTMLBody, "Issue1", Cells(3, 1))
            .HTMLBody = Replace(.HTMLBody, Lastfirstname1, Cells(3, 2))
            .Display
        End With
        Set MyItem = Nothing
        Set myOlApp = Nothing
        Rows("3:3").Delete Shift:=xlUp
    Wend

End Sub