NEW DELHI—The death of the leader of Tamil Nadu state, a former actress revered by many as Amma, or mother, heralds a period of uncertainty for foreign investors and a potential political opening for Prime Minister Narendra Modi.

Chief minister Jayaram Jayalalithaa, whose fervent following during a 35-year career in politics was matched by few other Indian politicians, died late Monday at age 68, after a long illness.

A close aide was quickly sworn in to succeed her, but he lacks her mass appeal in a political landscape built around personality cults. That could spell a bumpy transition and reshape politics in the southern state of 70 million people, where major auto companies including Ford Motor Co. and Hyundai Motor Co. have offices.

If her party “doesn’t manage the succession well, it could open up the field to other parties,” said E. Sridharan, academic director of the University of Pennsylvania’s Institute for the Advanced Study of India in New Delhi. Mr. Modi’s Bharatiya Janata Party “and others are among them.”

Ms. Jayalalithaa’s All India Anna Dravida Munnetra Kazhagam party emerged as the second-largest opposition force in India’s Parliament after national elections in 2014, narrowly trailing the Congress party, on the strength of its performance in Tamil Nadu.

It remains a regional party, however, trading power with a regional rival.

Mr. Modi’s BJP, meanwhile, has been seeking to spread its influence outside its traditional strongholds as it emerges as India’s dominant political force. It hasn’t had much electoral success in Tamil Nadu, but might sense an opportunity in the vacuum left behind.

“Her demise has left a huge void in Indian politics,” Mr. Modi tweeted.

A minister in her government, O. Panneerselvam, was sworn in early Tuesday as chief minister. Although her term runs until 2021, the party could replace him before then, but lacks a clear, charismatic successor. Ms. Jayalalithaa hadn’t groomed anyone.

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Jayaram Jayalalithaa, the much-loved chief minister of India’s Tamil Nadu state, in Chennai on May 23. PHOTO: AGENCE FRANCE-PRESSE/GETTY IMAGES

Any political instability or government paralysis is likely to worry the foreign investors that Ms. Jayalalithaa attracted through high-profile events showcasing her long-term economic agenda, which included multi-billion dollar infrastructure investments and power-generation projects.

“We are anxious because it is tough to predict what turn politics will take,” said Rafeeque Ahmed, who heads the Federation of Indian Chambers of Commerce and Industry in Tamil Nadu and runs a leather-export business.

He said government decision-making needed to pick up after being “in a state of coma” since Ms. Jayalalithaa was admitted to the hospital in September.

On Tuesday, thousands of her supporters lined up outside the building in the state capital of Chennai to view her body, which was draped with the Indian flag.

The daughter of an actress in Tamil-language films, Ms. Jayalalithaa’s first career was in show business. She was featured in more than 100 movies before turning to politics in the 1980s, serving as chief minister of her state four times.

She governed her party with absolute authority; it referred to her as the “Iron Lady of India” in a late-night tweet announcing her death.

Her popularity was undiminished by corruption investigations and a short imprisonment in 2014, after she was convicted on charges that she had amassed wealth well beyond her known sources of income. Local newspapers and her party at the time reported suicides and self-immolations by devastated fans.

The verdict was reversed by an appeals court and she returned as chief minister last year, leading supporters to celebrate by pouring milk on giant posters of her—a Hindu ritual usually reserved for deities.