Doing a little reading helps . . . and often gives more incentive to be growing our own food.
Potatoes must be the most sprayed (read "poisoned") crop on Earth. They don't like competition - so the fields are sprayed with herbicides. Pests can sweep thru the crop and defoliate the plants - so they are sprayed with insecticides. Nematodes attack the tubers underground - so the soil is often treated with chemicals to kill those critters.
After they are out of the ground - they can be sprayed dipped and fumigated and often are. Notice how sometimes a few of the spuds coming out of the bag from the store have blackened and cracked skin. Blessed be the food industry . . .
Potatoes are a cool season crop so warming the soil with tires shouldn't be of much help. Search the TheEasyGarden for what might be ending up in your soil from using old tires in your vegetable gardens. Just gather the soil around the plants as they grow in the open garden. It's the above-ground part of the plant that is "growing" the tubers underground. The starch is coming from sunlight and photosynthesis.
KeyLimePie, flowering is often used by gardeners as an indication that new potatoes are beginning to form. Some varieties don't flower or don't flower much but if yours are blooming - it may be time for a pot of creamed peas and new potatoes

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Using organic potatoes from the store should work if you want them for seed potatoes. Presumably they haven't been treated to inhibit sprouting.
Regarding sprout suppression: ". . . to further minimize sprouting, chemical treatments can be applied. Maleic hydrazide is sprayed onto the foliage 2-3 weeks after full bloom . . . Chemicals to suppress sprouting can be applied as a dip or aerosol treatment to tubers after harvest and after injuries are healed. Inhibitors should not be applied to tubers intended for seed use. "
Steve