WASHINGTON—The Library of Congress—long a sleepy, noncontroversial staple of the nation’s capital—has become the latest battleground in President Trump’s power struggle between the branches of government.

Lawmakers are raising separation-of-powers concerns as the White House seeks to install a Justice Department official at the legislative-branch institution.

Carla Hayden, the first African American and woman to lead the Library of Congress, was dismissed without explanation.