Policeman is a Firefox add-on which allows users to decide what type of content they'll allow on individual sites (think along the lines of RequestPolicy or NoScript).
At a minimum, you could use the add-on to find out which domains a site is accessing to download its content.
Next, you might add some rules to customise what's going on, allowing or blocking access to images, media, style, scripts, objects, frames and more, for both the originating and destination domains.
Setting this up can take a while, as the developer warns: "This add-on requires quite a lot of interaction from user to get things working. Some sites will not work properly until you have allowed the required content." (Fortunately you can temporarily disable any filtering, if necessary.)
But if you'd like full control over Firefox's actions - or you're just interested in seeing what websites are doing in the background - then Policeman gives you plenty of options.
Policeman gives you a vast amount of control over the content Firefox will access, but setting this up to work for general browsing could take quite some time.