Thanks. That helped me figured it out.

1) I had a vbCrLf that was adding the space above the header. Removed that
2) I needed to set the RowHeaderVisible to false. I was only thinking column header but guess you have two headers.

Code:
    Private Sub btnCopyAll_Click(sender As System.Object, e As System.EventArgs) Handles btnCopyAll.Click
        Dim sClipboard As String = String.Empty
        dgvResults.AutoResizeColumns()
        dgvResults.RowHeadersVisible = False
        dgvResults.SelectAll()

        dgvResults.ClipboardCopyMode = DataGridViewClipboardCopyMode.EnableAlwaysIncludeHeaderText

        If Me.dgvResults.GetCellCount(DataGridViewElementStates.Selected) > 0 Then
            Try
                ' Add the selection to the clipboard.
                Clipboard.SetDataObject(Me.dgvResults.GetClipboardContent())

                ' Replace the text box contents with the clipboard text.
                sClipboard = Clipboard.GetText()

            Catch ex As System.Runtime.InteropServices.ExternalException
                Throw New Exception(ex.Message, ex.InnerException)
                MessageBox.Show("The Clipboard could not be accessed.", "Please try again.", MessageBoxButtons.OK, MessageBoxIcon.Information)
            End Try

        End If

        Clipboard.SetText(sClipboard)

    End Sub