To remove text you first select it, programmatically by setting SelectionStart and -Length. Then you use the Cut Method:
richTextBox1.SelectionStart = 20;
richTextBox1.SelectionLength = 120;
richTextBox1.Cut();
If you want to avoid putting the removed text to the clipboard you can set it to ""
instead of 'cutting' it:
richTextBox1.SelectedText = ""