I'm having a similar problem when I watch Youtube. If I turn Youtube on the memory clock goes up to 1050MHz like normal but if I start up a game with Youtube running the GPU core clock doesn't increase past 300MHz. Other than that it's only Minecraft. I suspect it's an issue with OpenGL and the ATI drivers.
Edit: I had this exact problem with my old card, a 5750. I also have a friend who has a 5770 and a 6870 and he has the exact same problem.