WASHINGTON—The Trump administration has veered away from a rule of thumb in place at the White House and U.S. Treasury for roughly a quarter-century: Avoid commenting on the dollar, the Federal Reserve or daily moves in stock markets.

President Donald Trump illustrated the departure on Monday with a tweet taking note of recent currency volatility. “Russia and China are playing the Currency Devaluation game as the U.S. keeps raising interest rates,” he said. “Not acceptable!”

During...