WASHINGTON—The Treasury Department declined to turn over President Trump’s tax returns by a Tuesday deadline set by a key House Democrat, a decision that escalates a battle over the documents and could prompt a court fight.

Rep. Richard Neal (D., Mass.), chairman of the House Ways and Means Committee, this month made the formal request for several years’ worth of the president’s returns under a law that would require them to be handed over. In a letter Tuesday in response, Treasury Secretary Steven Mnuchin outlined constitutional...