Several public funds with holdings in Facebook Inc.,including New York City’s pension funds, are backing a shareholder proposal to push out Chief Executive Mark Zuckerberg as chairman of the social-media giant’s board of directors.

The state treasurers of Rhode Island, Illinois and Pennsylvania, as well as the New York City comptroller are joining the shareholder proposal previously filed by Trillium Asset Management in June, the officials said in a release on Wednesday.

The announcement has no practical effect on Mr. Zuckerberg and his position as both CEO and chairman because he has a lock on the bulk of Facebook’s supervoting shares, each of which gives him 10 times the votes of average shareholders. According to Facebook’s latest proxy, his share of the voting power among Facebook investors was 59.9%.

Still, the news heightens scrutiny of the company’s corporate governance after a string of missteps and controversies, and reflects a view among some investors that Mr. Zuckerberg needs additional support guiding the company.