Its difficult to say at this point without the full game out and just the beta to go on, I own several different computers at many different levels of performance and the beta (probably full game too) takes some fps drops in large fights, they are major drops on computers with weaker CPUs, notably dual cores. This is alleviated by turning physics from high to low. So when we talk about GPUs and "high settings" we should exclude the physics option as that's calculated by the CPU.
What card is the minimum for high settings depends on what you find as playable, some people are ok with going down to 30 fps, whereas people like myself need 50+ fps or it feels choppy. Because diablo 3 can have so much going on at once I think people under-estimate how much performance is needed, based on my experience with most of the ranges of gaming parts and the d3 beta, my recommendation for the cheapest smooth play for high settings would be a 6850 or equivalent for 1080p.