Never heard of Tire Taters up here in VT. Sounds like a great idea. I grow mine the old fashion way in hills. I know some folks that grow them under straw or hay. Only thing with that is the snakes. I hate snakes. 
We just finished up our last year Yukon Gold potatoes I dug up last Oct. This past weekend I put in a few rows of Yukon Gold for my early crop.
Snakes?
That is different.
I have never done the tire thing, (the sig. has fits, as it is, for me saving straw bales used on the roses plus I also have what looks like a grave where I buried the leaves that will not fit in the mulch pile because she will not accept that a garden cannot look like neatly organized front-room) but I have done deep burying; surface under straw, not good, hay or leaves, better; shallow buried under chopped leaves, which leaves a good semi-rotted mulch that when turned over by hand, makes the soil very nice in spring for tilling; trenches that are left open to the sun before planing to help control scab, and then filled slowly similar to the tire method, and a few others.
I have found that, with our soil, turning the garden over in the fall by hand, with a sand shovel, so it leaves a large chunk-lumpy finish with many air gaps, makes spring work much better and easier.
Of all the methods I tried, I have found that the quality of soil still is Number one in having a good potato crop.
