The proposal is largely symbolic, since Zuckerberg holds absolute control of the board. But it comes at a difficult time for Facebook, as security breaches plague the company and spur questions around corporate oversight.
"We need Facebook's insular boardroom to make a serious commitment to addressing real risks — reputational, regulatory, and the risk to our democracy — that impact the company, its share owners, and ultimately the hard-earned pensions of thousands of New York City workers," New York City Comptroller Scott Stringer said in a statement to CNBC.
Stringer joined a previous motion by Trillium Asset Management in calling for Zuckerberg to step down.
"An independent board chair is essential to moving Facebook forward from this mess, and to reestablish trust with Americans and investors alike," Stringer said.
Stringer oversees New York City's $160 billion pension fund. State treasurers for Rhode Island, Illinois and Pennsylvania are also joining the motion.
Similar motions to reclaim voting power have been made before, though none has passed. The latest proposal will be voted on at Facebook's next shareholder meeting next year.
A spokesperson for Facebook declined to comment.