If I actually headed due South, I'd run into North Carolina. That's not really possible because of the mountains, so the closest thing by road would be to take I-75. If I did that, I'd run into Georgia just south of Chattanooga.
If I drove due south from where I sit, I'd lose the car in the Straight of Georgia before I got to the US border. From home, due south would take me into the Fraser River fairly quickly, but minor deviations would allow me to use first a bridge and then a tunnel, thus saving the car, and from there the first border would be the US border -- into Point Roberts, WA.
I live at the very bottom of the North Island so if I were to head South I would be heading into the Cook Strait (3 hours on the boat) but if my car managed that I would be in a neat little seaside town called Picton