By far the weirdest thing I've ever seen. On some machines (!) IE (!!) attempts to load an SWF file via HEAD request instead of GET. This happens only when the page is running in Facebook IFRAME application - when running solo works fine. Naturally, on FF and Chrome everything works just fine. The SWF is embedded using jSWFObject. There's no visible common feature to the machines which experience this problem - laptops, desktops, XP, Vista, 7.
Any ideas?
It turns out that the issue was due to the SWF being hosted on an HTTP server, while the caller page was under HTTPS which wasn't properly configured.