I wouldn't do it.
I understand the urge because the ease of unearthing the tubers one they are done growing. But, potatoes don't actually like warm feet. Cooler soil may give you better results.
But mostly, tires are super toxic.
Here's a quote from a study testing toxicity from old tires to fish:
"Overlying water from both new and used tires was lethal to rainbow trout (Oncorhynchus mykiss) but leachate from used tires was more toxic (96-h LC50s - 11.8 to 19.3 %v/v) than leachate from new tires (96-h LC50s - 52.1 to 80.4 %v/v). In addition, leachate remained relatively toxic to rainbow trout over time (8 d for new and 32 d for used) after tires were removed from the aquaria indicating that the chemicals responsible for toxicity degrade slowly and are non-volatile. "
https://www.sciencedirect.com/science/article/pii/004565359390100J
I'd suggest the Ruth Stout system (or something like it).
Just as easy, better yields, and improves your soil rather than poisoning it.
I grow my spuds in raised beds made of cob, then layer straw over then as they grow upward. Very good results.