Hi AnjL,
While I haven't grown potatoes using stacked tires, I have grown them many times in 24" planters, wire cylinders lined with landscape fabric, as well as in-ground. Using this method, I have had the best success using a mix of cut straw, dry leaves, old potting soil & whatever compost is lying around.
I plant them into the first layer of about 6" of potting soil. As they grow, I use the aforementioned mix to add to the existing layer up to (but not over) the top leaves, adding as they continue to grow. Potatoes don't require a lot of nutrients, so I only fertilize a couple times. Remember, you're growing tubers.The mix stays loose, so watering is essential but -- be careful not to overwater. The mix will pack down somewhat, but not compact like soil does. Rain shouldn't be an issue unless we're getting crazy-rain. You can always cover temporarily.
The beauty of this method is no-dig. After they flower & the foliage dies back, you can reach in & harvest all or a few. You can certainly use a potting soil for growing this way. I was experimenting to see how the results of this method compared with in-ground potatoes, growing the same varieties using the vertical & in-ground methods.
I also grew them in an old trash bin that had the bottom removed. Harvest was achieved by tilting the trash bin, but I needed another person to help lift it. All the potatoes are accessible at once. This method of growing them "vertically" saves valuable garden space. I did not notice any difference in flavour or size. This is a bit late for your post, but I hope this helps!
