Elon Musk's erratic tweets could get the company in trouble, and it's about time someone stepped in, corporate communications expert Paul Argenti told CNBC on Tuesday.

"At some point somebody, like an adult, has to come in and say, 'Mr. Musk, you need to be more careful about your communications,'" Argenti, a professor at Dartmouth College's Tuck School of Business, said on CNBC's "Power Lunch."

That "adult" likely isn't Tesla's board, according to Argenti, who graded the board a "D at best" for its handling of Musk's recent erratic behavior and due to its tenuous independence.

"This is one of those things where you kind of look at it as almost a predictable surprise. When you have someone as loose a cannon as Elon Musk is, and given the kinds of communications he has been having, they probably should have done something prior to this to try to mitigate some of these losses that are coming in now — but they didn't," Argenti said.

Tesla has faced widespread scrutiny in the week since Musk tweeted that he was planning to take Tesla private when the stock reached $420 per share and that he had "funding secured." The Securities and Exchange Commission reportedly intensified inquiries into Tesla based on that tweet, which may have violated a rule that prohibits publicly traded companies from announcing plans to buy or sell securities if executives don't intend to follow through.

Tesla's board has faced criticism from both investors and advocates who question its independence and oversight of Musk. The electric car maker announced on Monday its board has formed a special committee, composed of three independent advisors, to evaluate Musk's desire to go private. The practice is not uncommon among companies mulling decisions that could present conflicts of interest.

The board's move sends a signal Tesla is seriously considering going private, but it does little to mitigate some of Musk's past, offhand remarks, management guru Jeffrey Sonnenfeld said.

"Right now if there were no board of directors, the company could not be doing any worse in terms of its misguided public interface," he said. "This is not just a public relations matter, it is an investor relations issue. A lot of people lost a lot of money from these comments, and we should be seeing those shareholder lawsuits launch any second — as they should."

As CEO of a public company, Musk "has made himself wealthy on other peoples' money. He has to be accountable to them," said Sonnenfeld, senior associate dean for leadership studies at Yale School of Management.

Sonnenfeld said shareholder lawsuits and an SEC crackdown could provide the incentive that Tesla's "passive" and "paralyzed" board needs to confront Musk.

"I think it's very likely they are going to get some sort of SEC citation. It won't be a huge financial burden but will tarnish the brand. And the shareholder lawsuits will give the board enough leverage to tell Musk off," Sonnenfeld said.

As to the consequences for Musk, Sonnenfeld isn't ruling out removal.

"It did Steve Jobs a lot of good to be forced out when he started creating a religion unto himself. And he came back much improved," he said.

Shares of Tesla closed down 2.46 percent at $347.64 per share on Tuesday. Tesla stock has lost 8.4 percent in the week since Musk's tweet on Aug. 7.

Tesla did not immediately respond to CNBC's request for comment.