As MoviePass has struggled, theater companies and other competitors have sensed a business opportunity and are moving quickly to satisfy the public’s newfound appetite for subscription movie tickets.

Last month, MoviePass limited its 3 million subscribers to three films a month, from as many as 31 previously, and restricted the number of titles available for tickets—while still charging $9.95 a month. The changes were made to stanch the financial bleeding of its earlier business model.

Meanwhile,...