I have this exact same problem.
It occurs all the time when I try to use crossfire (2x 7950) and about 70% of the time when I try to use 1 card.. I leave task manager open so I can see memory usage, it only gets up to 4GB when it crashes and reports low memory.
I have 8GB RAM, I5-3570K and 7950 crossfire. I'm using the latest 13.10 beta2 drivers.
When it does finally work, I'm getting 60-70fps on max settings, but with 2xMSAA, occassionally it dropped to 5 fps for no apparent reason.
My CPU usage is at 100% whether it's at 3.4GHz or 4.7GHz.
Personally, I would expect better from a public Beta release, Beta should be close to full release, only with minor bugs, but this game still has major issues. Considering AMD's 'exclusive deal' with DICE I'm amazed crossfire doesn't work.
Additionally, the graphics are poor. Compared to Crysis 3, which takes 2x7950 to get 60fps, BF4 only takes one and I can see why. Everying is so wooden and false. I was expecting better.