I don't use Firefox because it grinds my gears but even so, if you open a file it has to be saved somewhere. You can probably opt to choose a different destination but it'll have to be saved nonetheless. Browsers that don't appear to be saving files and appear to just let you open them are usually actually saving the file in a temporary folder somewhere. No matter what, it'll end up on your hard disk.
I imagine the Preferences would have an option to change where files are located when they're opened/saved. Have you looked through there? 🙂