By follow-tail I take it to mean that you want to continuously append text to the end of the textbox and have the textbox automatically scroll so that the new text is visible.
You've already set the text box to multiline, and I advise that you add vertical scrollbar if you haven't already.
The following code will move you to the bottom of the text box each time you add a line
TextBox1.Text += TweetText + System.Environment.NewLine
textBox1.SelectionStart = textBox1.Text.Length
textBox1.ScrollToCaret()
EDIT - I'll try that again in C# !!!
textBox1.Text += (TweetText) + System.Environment.NewLine;
textBox1.SelectionStart = textBox1.Text.Length;
textBox1.ScrollToCaret();
Where TweetText is a string