Starbucks Corp.'s board is pushing back against a coffee-cup recycling proposal from shareholders worried about ocean pollution.

The group of investors, led by shareholder advocacy nonprofit As You Sow, has submitted a proposal for the coffee chain’s upcoming annual meeting asking its board to report on worldwide efforts to recycle its plastic and paper cups. The board told investors to vote against the proposal in a Jan. 25 proxy filing, citing Starbucks’ commitment to help develop a new kind of...