Right now, I would not. I'm also rocking a 5850 but nothing is a big enough of a leap up to really invest in at this point. New stuff should be out relatively soon-ish.
Do you actually have a reason to upgrade the cooler? Unless you're OCing or the CPU is somehow hitting high temps (70C+), there's no need because its doing its job. Plus, you'd basically be disassembling the entire computer just to replace the cooler.
I'd agree with the others and just hang tight until you really see your game performance go down the drain. Sure, the new hardware is great, but why buy it if you won't see a difference?
Unless there's something the two HD5850s can't handle right now, don't bother. You'd be looking at considerable expenditure to get something that's noticeably better (i.e. HD6950CF or up, which is a good $500 initial outlay).
Dual 5850's OC'd to 800MHz GPU core speed is still a very fast setup even today. I have two game machines at home, one with dual 5850's and one with dual 6870's. The 6870's are faster at stock clocks but when the 5850's are OC'd, the 6870's can't match them except in games with a lot of tesselation. The 5850's were, and still are a great value.
I'm running a single XFX 5850 on a res of 4800x900; there is really nothing that chokes this thing up, save for maybe Metro 2033 or something like that maxed out. Was a solid purchase at the time, and right now, I'd say it's standing up pretty well against the 6xxx series. I'm gonna wait for BF3 to hit, but even once it does, I doubt I'm going to need to do much (although I'll still probably want to.)
It's all subjective, some people cannot stand running games without any AA, medium shadows and only 30-40fps, whereas to a lot of people they really don't care. How demanding a game is, is only relative within your expectations for graphical quality and frame rate,