HONG KONG—The city’s fast and efficient subway system has been the lifeblood of its nimble protest movement, ferrying demonstrators from one district to another for flash protests and helping demonstrators flee from police.

Now, Hong Kong’s subway operator says it will no longer tolerate protests inside its stations, which have increasingly become venues for vandalism by demonstrators seeking a safe haven amid standoffs or clashes with police. The move follows criticism of the company by Chinese state media for being soft...