Most of the time a video card is hardly doing any work, if however you were running it heavily for prolonged periods then this would for a card like the HD 7970 cause it to draw more than the official capacity of the cable and cause the cable and more importantly the very thin wires on the main Mac Pro logic board to heat up and perhaps fail. In my case I don't stress the card so it was a minimal risk, however if you were doing several hours of heavy gaming, or running a BitCoin mine, or some other similar stressful task it might be a risk.
On the Mac Pro the PCIe power leads go via the logic board whereas in many PCs they go direct to the power supply, this means the logic board is a potential point of failure if over stressed.
Two possible results would be either melting the logic board wires, or causing the Mac to shutdown as a preventive measure. Each 6pin connection is officially rated at 75w, like normal electrical cabling you can pull more than the official amount but this causes excess heating and potentially failure or even fire.
Note: The RX 480 apparently is only supported in Sierra so as you have a 2012 model which you can run Sierra on you will be ok.