PHOENIX — An excavator clawed away at a squat, battered building on the edge of downtown one morning, tearing the structure down in chunks that sounded like firecrackers as they crashed to the ground — heaps of discarded history in a city that prizes what is new.

From the other side of a chain-link fence, a cadre of preservationists watched with lament. The Madison Hotel — a boardinghouse for traders, travelers and tramps dating back to Arizona’s territorial days — was coming down unceremoniously to make way for a parking lot.

Its younger neighbor, the Hotel St. James, is to be next for the wrecking crew.

The hotels’ demolition permits were issued midsummer, though no one seemed to have heard about it until Michael Levine stumbled upon a lead-cleanup crew stepping out of the buildings sometime in August and thought to make a call to City Hall. Mr. Levine, 44, is an artist who has made it his business to buy and renovate some of the surviving buildings in Phoenix’s vanishing warehouse district, where, he said, “it’s been all about buying low, building cheap and selling out.”...