New tariffs intended to bolster the American steel and aluminum industries are starting to have the opposite effect in a key part of the U.S. supply chain.

U.S. steel producers are benefiting from tariffs that make it more expensive for companies to buy the metals overseas. But some U.S. firms that use the metals to make everything from refrigeration parts to wheels say the tariffs have led to higher materials prices that are forcing them to charge more for their products. These firms say that in some cases, customers are turning to foreign suppliers that use cheaper, tariff-free metals to make the same products they can then export to the U.S. without bumping up against the new trade barriers.

The fallout, while so far limited, illustrates how efforts to protect some U.S. companies can cause unintended pain for others.

“This is a nightmare for steel consumers,” said H.O. Woltz III, chief executive of Insteel Industries Inc., a North Carolina maker of concrete reinforcements. Mr. Woltz said some of Insteel’s customers have indicated they will boost imports.

The Trump administration imposed broad tariffs on imported steel and aluminum earlier this year, sparking fears among metal-consuming manufacturers that their customers would ditch them for suppliers abroad.