"I can't imagine any sound investor who has money in the company or any independent board member would want him to remain as CEO in light of recent performance," Lutz said on CNBC's "Closing Bell."
The New York Times published an extended interview with Musk on Friday in which he said the past year has been "excruciating" and "the most difficult and painful" of his career. The interview follows months of erratic behavior on Musk's part, both on and off social media. Most recently, the CEO tweeted that he would take Tesla private at $420 per share and had "funding secured," which has invited scrutiny from the Securities and Exchange Commission.
Musk took to Twitter in July to call a British cave diver who assisted in the rescue of a Thai boys soccer team a "pedo guy." During Tesla's first-quarter earnings call in May, Musk dissed analysts, cutting off Sanford Bernstein's Toni Sacconaghi because of what he called a "boring, bonehead" question. Musk later apologized to Sacconaghi and to the diver, Vernon Unsworth, for his comments.
"Elon is tired, he's worn out. He's obviously got some emotional problems. He's self medicating. He has shown some disturbing signs of being somewhat volatile and unstable," Lutz said. "I think the right solution for Tesla at this point is to move him aside from day-to-day operation."
Lutz, an automotive industry veteran who has also served in top roles at BMW, Chrysler and Ford, has been a huge critic of Tesla in the past. In November of last year he said Tesla is "going out of business." Although he wasn't quite as critical this time around, he did raise some of the same issues, saying Tesla was "bleeding" profitability and "will probably have to go back to the capital markets for more money."
"In my personal judgment, the board should take action and find a CEO. Not get rid of Elon — keep him as the visionary, keep him as the titular head of the company, and give him the honor and respect the founder of the company deserves," Lutz said. "But that company needs professional management, and it needs it now."
J.P. Eggers, associate professor at New York University's Stern School of Business, agreed, saying it is likely there are many other things Musk would rather spend his time on than Model 3 production goals.
"We see this all the time with ... start-up founders or early leaders in these firms, where what they really want to do is do the vision, do the growth, build the ... reputation of the company. And when it comes to actually executing on the vision, they aren't always the best ones for that," Eggers said Friday on "Closing Bell."
As much as it might be better for Tesla — and its stock price — were Musk to step aside as CEO, Eggers said, he's not so sure Musk would be willing to stay on with the company in a secondary role.
"I have a hard time seeing him doing anything other than being completely involved or walking completely away. He's tenacious; that's what's made him successful to this point," Eggers said.