WASHINGTON—President Trump has enjoyed rare bipartisan cheering for his brass-knuckle trade fight with China, but that support will be tested as U.S. businesses and consumers shoulder higher tariffs on Chinese imports.

The yearlong conflict has barely dented the U.S. economy so far, but new 25% tariffs imposed Friday, along with the administration’s steps to expand levies to virtually everything imported from China, raise concerns even among some Republicans.

“The...