You know, I too thought about mentioning this, but then considered you may have run across something I didn't know.
potatoes & sweet
potatoes are related in that they're plants, but I think that's as far as it goes.
DO NOT CUT THE LEAVES!! It's the leaves that are converting the energy of the sun into sugars and starches that are eventually stored in the sweet
potato.
This is my first year with sweet
potatoes so I'm no expert. They will vine (mine are going like gangbusters) and they will root along the vine. I don't see anything wrong with leaving them in the tires, but don't continue to hill them up like your red
potatoes. Treat the tires like a container-- my sources say they do just fine in containers. They'll probably spread out over the tire, but that's no big deal.
Are you familiar with sweet
potato vine-- the purplish leafy vine that lots of people plant in their big
flower pots? That's the same thing as the sweet
potato we eat, just a different variety. (In fact, you can eat them, too!) If you are, that should give you an idea of how they will grow. Leaving them in your tire container would also cut down on the rooting along the vine.
But again-- I'm no expert! WOnder what others think.