Will this build run BF3?

System listed would do good for BF 3, but I am not really sure it could go to maximum settings, but maybe close, depending on your resolution, and DirectX version used. But as you said, you'd do better with the 6950, and maybe 6GB RAM instead of 4 as listed.

That's fine. I noticed that the author didn't include a cooler in his build. So stock cooler would be sufficient for this? Also, same thing with the 600W PSU?

There's no telling on how the game would run, as there are no benchmarks for it (apart from alpha), and the game isn't even released yet.
But since the recommended requirements for the game, says that it needs at least a GTX 460 or a HD 6850( not official though), so I'd say it can max out the game, with the resolution toned down, yes.

The HD 6950 would do even better, so if you're willing to get that, then it'll be better.  

My Phenom II X4 965 came with its own stock cooler and that is more than enough if you don't plan to overclock your CPU. Under load, my CPU temps are around 60 Celsius degrees. You should check if that CPU comes with its own cooler or you'll need an aftermarket one. The 600W PSU should be enough, if you want to go with a 650 or 700W one, you'll be more than comfortable in that department.

http://gamingrig.blogspot.com/2011/09/800-gaming-pc-sep...

Just for comparison sake, how about this build?

The abovementioned system is just as good, so you're a winner with either of the two.

GTX 560 Ti performs similar to a HD 6950, so both are equal in terms of performance.  

In that case the first deal might be better because it's cheaper. What about the CPU? I've heard a lot about the NVIDIA s5i 2500k. Isn't the CPU from the second build better than the first?

They're in the same price range, and even you might find a better deal for AMD CPU here. Also, read this article too, you'll see how close the performance of the two CPU's are.

Yes, the CPU from the second build is better than the first.
AMD CPU's are cheap, but an i5 2500k is much better.

If you're sticking with the first build and you're only going to get better video card and CPU, then go for it.
You're going to have to change the PSU as well, if you're going for a HD 6950 or a GTX 560 Ti.

So to clarify, you're saying that for the first build I'm better off getting a better CPU in addition to picking up the HD6950? What if I upgrade just the video card to the HD6950? Do I still need to upgrade the PSU?

Yes.

A Corsair CX 500W PSU won't be enough to power a HD 6950, so yeah, you're going to have to change your PSU as well.