How can I run Firefox on a non-local virtual server?

I need to run Firefox (or some other web browser, but I prefer Firefox) on a non-local (i.e. not on my PC, not reachable via LAN) Unix virtual machine and then control it using keyboard and mouse, because some websites only work with Javascript and don't allow downloads unless I'm logged in.

-X

doesn't cut it, because it is a pain to use unless you are connected to the Internet with 10 Gbit/s. And I don't really feel like installing one of the fat "lightweight" desktop environments like Gnome or Xfce. And of course Firefox is not supposed to run as root.