You can add a bit of code to test for CutCopyMode, and exit the
procedure if it's true:

Private Sub Worksheet_SelectionChange(ByVal Target As Range)
Dim str As String
Dim cboTemp As OLEObject
Dim ws As Worksheet
Set ws = ActiveSheet

Application.EnableEvents = False
Application.ScreenUpdating = False

If Application.CutCopyMode Then
  GoTo errHandler
End If

Set cboTemp = ws.OLEObjects("TempCombo")
  On Error Resume Next
  With cboTemp
    .Top = 10
    .Left = 10
    .Width = 0
    .ListFillRange = ""
    .LinkedCell = ""
    .Visible = False
    .Value = ""
  End With

errHandler:
  Application.ScreenUpdating = True
  Application.EnableEvents = True
  Exit Sub

End Sub