May be the most serious threat that an attacker can do with TRACE is to access document.cookie property in order to hijack the victim's session. TRACE won't work in any browser (luckily).
For the case you described, it depends on the security layers implemented on the server the attacker runs it against and the question, as it is asked, is too broad to answer exactly. But put in your mind that the attacker will get information on how the server responds to a given request.
For curiosity, run this on Terminal (216.58.209.100= www.google.com
):
curl -X TRACE 216.58.209.100
You will get this error message:
Error 405 (Method Not Allowed)!!1