The Dawn Project recently published a full-page advert in the New York Times highlighting what it claims to be life-threatening danger to child pedestrians posed by Tesla's self-driving feature.
It reports that tests carried out by the safety campaigners reveal the car manufacturer's software would repeatedly run down child-sized mannequins in its path, as shown in this video demonstration.
In August, around a month before Musk's $44 billion takeover of Twitter, his other company Tesla demanded the Dawn Project remove its videos.
The firm sent a cease-and-desist letter and describing the clips as defamatory and misrepresentative of Tesla vehicles' capabilities.
That didn't stop the group from paying for its full-page advert in the NYT this month, but the project's founder Dan O'Dowd has subsequently been banned from promoting it on Twitter via an ad campaign.
In an email received by The Dawn Project on Monday, Twitter said it would not allow the promotion on its platform as it fell foul of its policy regarding “political” advertising.
Musk, who has recently reaffirmed his commitment to free speech, completed his long-awaited purchase of Twitter late last month and has further announced reforms to the social media giant's moderation policies.
He has told of his vision to create a "digital town square", having previously claimed Twitter had a problem with censorship and a left-wing bias.
"The move to prohibit the advertising of content criticising Musk's Tesla Full Self-Driving software raises serious questions over his commitment to free speech", O'Dowd said.
He added: "Claiming our safety tests, which raise awareness of a vital public safety issue, constitute political advertising is a serious incursion on free speech.
"The public must be made aware of the potentially lethal threat Tesla Full Self-Driving poses to child pedestrians, as our safety tests reveal. Perhaps all criticism of Elon will now be banned by Twitter!"
On its website, the Dawn Project, says its mission is to make "computers safe for humanity", while demanding that software "never fails and can't be hacked".
It says O’Dowd created secure operating systems for projects including Boeing’s 787s, Lockheed Martin’s F-35 Fighter Jets, the Boeing B1-B Intercontinental Nuclear Bomber, and NASA’s Orion Crew Exploration Vehicle.
Tesla is currently facing a criminal probe by the US Department of Justice over claims of "full self-driving capabilities".
The investigation, which could ultimately trigger criminal charges against Tesla executives, is centred on whether that claim constitutes defrauding consumers.
Yahoo News UK has approached Twitter for comment.