heavy metals are mostly an issue when the soil ph falls really low since heavy metals are not sought actively by plants and must be dissolved in the soil solution to be forced into the plant
but i would think about all the other junk in the tires, and also the idea that tires are really recyclable now, so using them for potato farming doesn't make much sense in a conservation sense
and the method i heard that seemed like a real winner was to have four stakes about 4 feet or so out of the ground, build your box (4x4?) fastening to the posts at the corners --1x 8s or 1x6s, or whatever
then the next layer just put on two boards on opposite sides of the box
the next layer do the other two sides,
and at each level pull the greenery to the outside edge of the box so it can grow out and up and in the end you have a tall box that you can't see for the greenery
the trick is to get as much leaf area as possible, since it is the photosynthesis producing the starch for the potato and for the continued growth of the plant
so if you do go ahead and use the tires i'd put a spacer between them,
enough to hold the soil, but still let the foliage out to the light