Hey Chrisc!
I think this problem is not related to WPF or WebBrowser control. If you sniff traffic from YouTube you find that content-type is application/x-shockwave-flash. It looks like IE reacts on this content type with Save file dialog.
But what you can do is create a standalone html and refer it to the video you are trying to show. Something like this:
<html>
<head>
<title>YouTube Video</title>
</head>
<body>
<object height="100%" width="100%">
<param name="movie" value="http://www.youtube.com/v/8VQ4f22-SeE&hl=ru&fs=1&">
</param><param name="allowFullScreen" value="true">
</param><param name="allowscriptaccess" value="always">
</param>
<embed src="http://www.youtube.com/v/8VQ4f22-SeE&hl=ru&fs=1&" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="100%" height="100%">
</embed>
</object>
</body>
</html>
Note, original video from your post cannot be embeded, because of permissions.
PS: To see the actual traffic I'm using FireFox + FireBug.