[Karsh]

Код:

procedure TForm1.Button1Click(Sender: TObject);
var
  i, x: Integer;
begin
  x:= 0;
  for i:= 0 to ListBox1.Items.Count - 1 do
    Inc(x, StrToInt(ListBox1.Items[i]));
  ShowMessage('Сумма = ' + IntToStr(x));
end;