I finally got the time to spend a half hour comparing the monitor and the video card, and the TV and the video card. Everything set to 60Hz should worth without tearing as long as V-Sync is on. So I think something or some somethings is blocking v-sync from actually being on. And I think the video card is feeding your monitor at rates that exceed the 60Hz rate on the monitor. Unfortunately, I cannot dig through your system to try to figure out what it is that is doing this. I don't even know what you have installed, but even if I did, I would not be able to see the settings.
But it is either a software setting, or malfunctioning hardware. You have the Sapphire, and the onboard video on your CPU. Try running the DVD off of the CPU's DVI port and see what happens. If The tearing stops, your video might be the problem. If it happens with the onboard DVI port, the monitor might be the problem. Or as I said earlier, it might well be a software setting somewhere.
Its tedious to do this, but sometimes, it the only way to figure out what is causing this. From what you have told me, you have everything right as far as the hardware goes.
Swap as many pieces as you can, 1 change at a time. Write down what you changed, and what the result as. Eventually, you will figure out where the problem is. And then you will either laugh or cry at how simple the fix was.
If there is anything else I can help with, just post here, and I will do my best.