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...