If you don't clean what the user puts in the textbox and submits, then yes, there is a chance for harm to be done.
You might want to check out the Microsoft Anti-Cross Site Scripting Library, as it is designed to help developers prevent just such attacks.
Also worth taking a look at is OWASP's Cross-site Scripting (XSS)
You might want to look into HttpUtility.HtmlEncode and HttpUtility.HtmlDecode as well. I just wrote a quick test, and it looks like it might address your concern in the comment below (about how to display the data to other users in the right format):
string htmlString = "<b>This is a test string</b><script>alert(\"alert!\")</script> and some other text with markup <ol><li>1234235</li></ol>";
string encodedString = HttpUtility.HtmlEncode(htmlString);
// result = <b>This is a test string</b><script>alert("alert!")</script> and some other text with markup <ol><li>1234235</li></ol>
string decodedString = HttpUtility.HtmlDecode(encodedString);
// result = <b>This is a test string</b><script>alert("alert!")</script> and some other text with markup <ol><li>1234235</li></ol>
ASP.NET Controls and HTMLEncode
I was going to post the information I had from my class, but I found a link that lists the exact same thing (for 1.1 and 2.0), so I'll post the link for easier reference. You can probably get more information on a specific control not listed (or 3.0/3.5/4.0 versions if they've changed) by looking on MSDN, but this should serve as a quick start guide for you, at least. Let me know if you need more information and I'll see what I can find.
ASP.NET Controls Default HTML Encoding
Here's a more comprehensive list from one of the MSDN blogs: Which ASP.NET Controls Automatically Encodes?