There is no way to actually know what would happen (primarily because we cannot know whether someone will actually sue), and a law might be broken with impunity (the violation could be ignored). We can at least say (somewhat) what the legal situation is, esp. what would constitute a violation of the law. To get to the bottom line, if there is any legal recourse, an aggrieved party would have to sue POTUS, and win the suit (compelling the president to do something different); or, the president could be impeached.
The War Powers Resolution of 1973, statutorily encoded in ch. 33 of Title 50 limits the president's authority to engage in military action. First, Congress must be notified within 48 hours of the action, second, armed forces cannot remain more than 60 days (plus 30 days for withdrawal) without a declaration of war, or congressional authorization (such as took place in 2001, and currently exists). In one instance, Clinton continued bombing in Kosovo two weeks beyond the 60 day limit, but the administration maintained that because Congress approved funding for the operation, they had implicitly approved the operation. There was a lawsuit by Congress, but the court ruled that the complainants did not have standing. The court also found that
Congress certainly could have passed a law forbidding the use of U.S.
forces in the Yugoslav campaign
and indeed there was such a measure, but it was defeated. The opinion continues:
Congress always retains appropriations authority and could have cut
off funds for the American role in the conflict
and again, that was attempted, but failed. The court found that
Congress has a broad range of legislative authority it can use to stop
a President's war making, see generally John C. Yoo, The Continuation
of Politics by Other Means: The Original Understanding of War Powers,
84 CAL. L. REV. 167 (1996)
so if suitable veto-proof legislation is passed, then a military action could be stopped (by specifically prohibiting use of US funds for military action).
Or, at least it seems that way. 31 USC 1341 forbids an officer of the government from making or authorizing expenditures or obligations unless authorized by law. Consequently, when Congress fails to pass a budget, government operations must cease. This is related to Art. 1 Sect 9 Cl. 7 of the Constitution, which says
No money shall be drawn from the treasury, but in consequence of
appropriations made by law
The aforementioned statute does not allow an officer to create an obligation: it does not prohibit an officer from doing a thing, which would ordinarily imply an expenditure or obligation. Notice that while previous federal shutdowns did result in unpaid furloughs to civilian workers, it did not completely close down the military. The military (accountants) explains
In case of a potential government shutdown, the Department of Defense
has no legal authority to pay any personnel - military or civilian -
for the days during which the government is shut down...
Military members cannot be paid for duty performed after the
expiration of the current CRA on September 30th. Once a CRA or an
appropriations act is signed into law, normal disbursement of military
pay will resume.
(Congress could of course authorize back pay when it's all sorted out, and they did). If Congress were to specifically prohibit any expenditure of US funds for action against North Korea, that might simply mean that soldiers would be ordered to work without pay.
It is unquestionably within the power of the president to order soldiers into combat. I don't believe that there has been a case pertaining to the power of the president to compel soldiers to fight if they are not paid, but it is pretty much unimaginable that the courts would hold that the president lacks authority to order soldiers to work without pay. Also, you can't just quit the military: it's not at-will employment. Refusing a lawful order is a crime under UCMJ (which is written by the commander-in-chief), and nothing makes it unlawful to give an order to solder if they are not being paid.
All that said, if there is implemented veto-proof opposition to military action against North Korea, there is almost certainly the number of votes needed to impeach. An impeachment is not subject to any judicial review whatsoever, so a president who went against such a strongly-supported no-funding law would likely also get the required 2/3 vote to remove.
