Jinling Road, marked by verandas that shield its sidewalks and display windows of violins and pianos, is a distinctly historical street in modern Shanghai. Now there are questions about its future.

“Reforming the old district is like a spring wind that can warm the hearts of thousands of households,” reads one of the ominous red banners hung recently along the eastern end of Jinling Road. Dozens of similar slogans tied to pillars that date to the early-20th century call on residents and businesses to cooperate with what appears...