WASHINGTON—President Donald Trump warned Harley-Davidson Inc. that moving production overseas would mark “the beginning of the end” of the iconic brand, a day after the manufacturer said it planned to make more motorcycles abroad to avoid European Union tariffs.

“A Harley-Davidson should never be built in another country—never! Their employees and customers are already very angry at them,” Mr. Trump wrote in a series of tweets Tuesday morning. “If they move, watch, it will be the beginning of the end—they surrendered, they...