Worst case scenario the neurons in that particular region would die. If the current stimulus was even higher then that local region of the brain would die and the person would no longer be able to sense his arm, or finger; but those sensations would likely regenerate because the brain is very adaptable.

This technology is clearly very cool, but there's two major hurdles to overcome before everyone's running around with one of these.

1. Controlling the device. Currently scientists/doctors control these brain computer interfaces (BCIs) by implanting electrodes into the patient's brain and finding neurons which code for particular movements (arm up or ring finger down). As the output device gets more complicated, like the arm here, doctors need to find more and more neurons to represent each degree of freedom of the output device.

2. Quality of neural recordings degrade with time. The current shelf life of the electrode arrays used in these experiments is ~1-2 years because after implantation, the brain's immune system rejects the device and neurons which code useful information die or move elsewhere. http://en.wikipedia.org/wiki/Chronic_electrode_implants