Sen. Elizabeth Warren sought to turn Facebook Inc.’s political ad policy against it, running an ad with a false claim about Chief Executive Mark Zuckerberg to highlight complaints about the social-media giant’s handling of misinformation.

The ad, which the Warren presidential campaign ran on Facebook, says Mr. Zuckerberg endorsed President Trump’s re-election, and shows an image of the two men shaking hands. The ad’s text then quickly acknowledges that the endorsement claim is false, and criticizes Facebook’s policy, announced...