CHICAGO—The Obama re-election campaign has quietly opened a counteroffensive against Republican-backed changes to election laws that Democrats say will suppress votes for their candidates and limit their get-out-the-vote drives. The effort, led by former White House counsel Robert Bauer, prompted the suspension of an Ohio law limiting early voting. Campaign officials produced educational materials to counter a Wisconsin law that requires voters to produce photo I.D.s—but disallows those used by Wisconsin colleges. By this spring, the Obama re-election campaign will mount what Mr. Bauer called an unprecedented "voter protection" effort, fielding thousands of volunteers in battleground states to help navigate...