I've just tried with fast random typing too, but still can't replicate it.
To see where that "<html>" is coming from, in Firefox for example you can open Firebug, go to Console tab, click on the latest GET request and then look at the Response tab. This should give you some hint.
If you want to post here this Response PAY ATTENTION it is just html and DO NOT post also some sensitive data from other tabs like PHPSESSID or SMFCookie that could bring you security problems (if it happens then just logout, clear all cookies and login again ASAP; it should be enough).
Maybe it's worth mentioning that it's not sure that this kind of bug - if it's a bug - would be considered prioritary and then addressed because of the new forum software in development.
But that's just my impression; let's see if there is some news from the staff.
Good suggestion. Found that it happens whenever I get a 503 error on the autocompletion request. The response is then:
<html>
<head><title>503 Service Temporarily Unavailable</title></head>
<body bgcolor="white">
<center><h1>503 Service Temporarily Unavailable</h1></center>
<hr><center>nginx</center>
</body>
</html>
Truncated then to the first line which replaces the nick I'm typing (didn't check the javascript part but it must be that).
Still, I don't know why I often get this with a good connection on two different browsers and computers, in different locations, and that nobody else can reproduce.
Indeed it's not high priority anyway.
