I hadn't changed or updated anything addon-wise. I had also tried FTP, with the same result. I'll give it a go again and also double check to make sure that something isn't blocked due to an updated filter list.
Update: FTP gave the same result. I did give it a go in IE and it worked fine. I'll fiddle with Firefox and let you know what I find. Thanks for the help!
Update 2/Problem solved: Tried it with all addons disabled, no change. Updated Firefox, same issues, noticed that the tab bar was showing even though it's set to hide the tab bar with only one tab open. Updated "Tab Mix Plus", finally got to the error page about an incorrect referrer. After clicking to add an exception and testing it with multiple files, I can confirm that I am now able to download via Firefox.