They called the judge’s refusal to halt the case while they appealed his rulings “a manifest abuse of discretion.” And they asked the appeals court to step in before Jan. 3, the deadline to reply to at least some of the subpoenas, to prevent “wide-ranging discovery into his personal finances and the official actions of his administration.”

In a statement, Karl A. Racine, the attorney general for the District of Columbia, said, “President Trump is going to extraordinary lengths to try to stop us from gathering information about how he is illegally profiting from the presidency. Unless the Fourth Circuit rules otherwise, we will continue to work hard to gather the evidence needed to put a stop to the president’s ongoing violations of the Constitution.”

The Justice Department is arguing that the framers of the Constitution sought only to prevent federal officials from acting as paid employees or officers of foreign or domestic governments, not from profiting from businesses that serve them. In any case, the department’s lawyers contend, the case is far too important to treat like an ordinary federal lawsuit.

Typically, appeals are restricted until after lower court proceedings finish. But the department noted that the 11th Circuit Court of Appeals had stepped in on an emergency basis in a case involving asylum claims of refugees. “The president is entitled to at least as much judicial solicitude in obtaining appellate review of threshold legal defenses” in this case, they wrote.

The lawsuit contends that foreign, state and federal officials deliberately book rooms at the Trump International Hotel and patronize its restaurant to curry favor with the president.

It alleges that since Mr. Trump was elected, the hotel has hosted events by the embassies of Kuwait, Bahrain and the Philippines and collected at least $270,000 from the government of Saudi Arabia. Other patrons have included a Malaysian government delegation and the governor of Maine, who was in Washington on business with the White House.

Among other entities, the plaintiffs are seeking documents from the Treasury Department, which was supposed to collect the hotel’s profits from business from foreign governments; the General Services Administration, which leased the property to Mr. Trump’s company; and 18 hotels, convention centers and other enterprises that compete for customers with Mr. Trump’s property.