First thing I tried in order to verify this was to disable FlashGot and see what happened. As a result, every time I clicked on a .dmg Firefox just wouldn't do anything. The little blue circle would go round and round, but beyond that, nothing. I tried editing the options in Speed Download to explicitly stop it from automatically downloading any file types, but this had no effect. I tried editing the list of file associations within Firefox, which did nothing, so I then tried following the instructions here on resetting that aspect of Firefox. This, similarly, did nothing to solve the problem.
Eventually I decided to search to see if anyone had had any luck with getting Speed Download to stop intercepting files on their Mac. I uncovered a rather odd forum thread regarding Speed Download integration with Growl. As it turns out, Speed Download's faulty Growl implementation was creating a "Recovered Files" folder in your Trash every time you log in - I had noticed this but assumed it was a Mac thing rather than a bug in an app I was running. According to the thread, this stopped happening with Mac OS X 10.6.7 - given that I haven't noticed it in quite a while, this would make sense - and neither the developers of Growl nor Speed Download were able to work out why it was going wrong. At least it's stopped happening now.
Anyway. That same thread also mentioned having trouble getting Speed Download to stop intercepting Firefox downloads, but because everybody had focused on Growl, there was no solution to be had. The person in that thread had even attempted to completely remove Speed Download but that hadn't solved his problem. However, I then found this article, in which someone gave Speed Download a go and decided to uninstall it afterwards. The author, who is my favourite person of the day, noted that to completely remove the app, it was necessary to go to "/Library/Internet Plug-Ins" (note that that's /Library, not ~/Library) and remove the Speed Download plugin from that folder.
Since I did that, Firefox's method of dealing with .dmg files has been restored to normal and I'm not having any more problems. Even putting the plugin back in the appropriate folder after a couple of browser restarts didn't bring the issue back. No idea why this didn't happen on my MacBook, but at least I've managed to fix it!
The reason I'm boring you all with this is because nobody else had actually written anything that answered my question online, so I want to codify it and make sure that there's some chance anyone else in my situation will be able to follow this instead of having to work it out themselves. So, in conclusion, if you're running Firefox with NoScript and Speed Download, and you're seeing white pages when trying to get .dmg files, try moving the Speed Download plugin, restarting the browser, moving it back in and restarting again.
(If I ever get around to making my website any good, guides to fixing problems that I've had will probably occupy a large portion of it - I've learned to do so many things on my Mac by simply Googling until my eyes bleed...)