A group of Tesla Inc. shareholders, including several U.S. state investment officials, called on the company’s board to make sweeping changes to its governance to enhance oversight of Chief Executive Elon Musk.

In a letter sent Thursday to three Tesla directors, the shareholders said Tesla’s board should create a plan to refresh its membership and to set time lines for some directors to leave, among other demands. The letter also said the board should permanently separate the positions of chairman and chief executive, both...