Why? To make an FPGA do what you want you need to be able to reconfigure it. If you have reconfiguration capability you need to have remote code execution. And in that case you have already lost.
As in, the FPGA would have to be carefully segmented so the accelerator couldn't be used to access memory it shouldn't have access to.
I don't think it would happen in a general purpose chip but I could see it happening in a smaller one like the exploits christopher Domas demonstrated against some embedded X86 cores.
Why though? Your Integrated Intel or AMD GPU can also access all of your memory. I don't see how an FPGA provides any additional attack vector. As I said you'd need code execution privileges anyway and once you have that your system is already owned.