Is there a way to completely disable cookies from the Tornado server?  I'm not using them for my public-facing site and I don't want to have the annoying pop-up to get European compliance.
The only time Tornado sets cookies on its own is if you use the xsrf_cookies application setting. If you turn that off and there are no other calls to set_cookie or set_secure_cookie in your application, no cookies will be set. 
XSRF is a serious security issue and you should carefully consider whether you need an alternative solution if you don't use xsrf_cookies. However, if you don't have any other cookies (or equivalent forms of authentication), you're probably OK.
Turn off cookies in Tornado