Since nobody else has posted this yet here, it has been confirmed that post processing is not working with Radeon 5-series of video cards (or at least 5870). 4-series and earlier still work as usual. If post processing is enabled and you have 5-series Radeon, game will crash to desktop when a mission is loading. This is probably Catalyst driver bug.
Actually. . . I have a mantis report in on this very same issue, although I own a Radeon 5830. Mantis issue is 0002204 - I have tried Catalyst 10.3-10.6 and the issue occurs with 3.6.12-RC2 and 3.6.12-RC3.
Interestingly enough a developer who was working with me on this built me the (then) latest trunk build in 3.6.13, and Post Processing worked fine. The fix is in the trunk code but has not been back ported to the 3.6.12 code base. Comment above by Chief indicates to me it will likely not be, either.

I probably don't have the most recent drivers, and knowing these kinds of bugs I'm not exactly pumped about upgrading drivers anyway.
Do the 57xx and 58xx series have the same drivers?
Yes, all ATI Radeon 2xxx, 3xxx, 4xxx and 5xxx series cards use the same exact drivers, and ATI releases a new driver each and every month. Current driver is 10.6, meaning:
(20)10.6(June)
It might also be the driver version you're using. It seems that ATi has been throwing out some pretty weird tweaks that are bugging a few select things in the latest drivers. I wouldn't be surprised if something similar was preventing post-processing from working.
The 10.3 release I was using when I posted the issue in Mantis was the OpenGL 4.0 Preview driver. Catalyst 10.4 and 10.5 used the bog-standard ATI OpenGL driver. I did a clean driver install between each version to avoid problems while I was testing.
Catalyst 10.6 (current) is the official release of the new ATI OpenGL 4.0 driver (for 5xxx series cards. It is 3.something for older cards).
30% of steam users is far closer than real market share to FSO current target audience.
Yes, but again I would respectfully argue that FSO users, by nature, are more "enthusiast" users in general. I mean, it's not like FSO and the mediaVPs are any sort of easy for the novice computer user to install and run here

. My thinking would still be that the few users that both browse this site AND that don't understand the whole SSE thing are more likely to be Intel owners anyway. Saying "Newer than Pentium 4" is probably still insufficient. Perhaps "for Pentium 4, Core and i-series CPUs and equivalent" would be better. After all - the purpose is to help avoid the same question coming up again and again in the forum, not to educate people on the finer points of the various SSE instruction sets and supporting hardware.

Remember - bottom up, not top down.
Post merge for great justice. --Jeff Vader