update textbox if not empty

Just surround the CommandText bits with a test on the contents of the text box ...

If Trim(TextBox8.Text).Length > 0 Then
     SavInto.CommandText = "UPDATE Table1 SET Place = '" & Trim(ComboBox2.SelectedItem) & "' WHERE FilNum ='" & (TextBox8.Text) & "'"
End If

Note that only your last command will actually be executed as you're overwriting SavInto.CommandText each time.

Finally, do some research on SQL Injection and replace the string concatenation with proper parameters ... http://msdn.microsoft.com/en-us/library/system.data.oledb.oledbcommand.parameters.aspx[^]

[Edit - an alternative approach that is extensible, avoids SQL injection and uses a single update statement]

Dim tbList As StringBuilder = New StringBuilder("(")    'More efficient than string concatenation
        For Each t As Control In Me.GroupBox1.Controls          'I placed all the relevent TextBoxes into an (invisible) group box
            If TypeOf (t) Is TextBox Then
                If Trim(t.Text).Length > 0 Then                 'Only interested in textboxes with content
                    tbList.Append(Trim("'" & t.Text) & "',")    'Build up a list of the textbox contents
                End If
            End If
        Next
        If tbList.Length > 1 Then   'Effectively checks for all boxes being empty
            Dim cmdList As String = tbList.ToString()
            If cmdList.EndsWith(",") Then cmdList = Strings.Left(cmdList, cmdList.Length - 1) 'Remove last comma
            cmdList += ")"          'Put the closing bracket in place
            SavInto.CommandText = "UPDATE Table1 SET Place = ? WHERE FilNum in " & cmdList
            SavInto.Parameters.Add(ComboBox2.SelectedItem)  'Use parameters to avoid SQL Injection
            Conn.Open()
            SavInto.ExecuteNonQuery()       'Execute the single query
            Conn.Close()
        End If