Mr. Musk’s involvement in politics and his financial support for the president’s campaign were once seen by investors as a benefit to Tesla, fueling a steep rise in company shares after the election last year. That association has now turned into a liability. Tesla shares are down 40 percent from their peak in December.

Longtime supporters of Tesla and Elon Musk called on its board to pull the CEO back to the troubled EV maker after he said he is starting a new America Party.