The president cannot, in the exercise of his oath of office, ignore it. It is the Law of the United States and his duty as president is to "faithfully execute the laws of the United States." If he believes that the law is unconstitutional he cannot unilaterally ignore it. Per his oath of office he must either obey it, or challenge it in court.
To claim that it does not apply because the United States is not engaged in "hostilities" when he is ordering our troops to drop bombs on people is to remove the legitimate basis for the actions. If our troops are not engaged in formal "hostilities" then they are engaged in acts of murder.
If Obama ignores this law, he should be impeached. His duty, if he believes it is unconstitutional is to direct the Solicitor General to file a suit before the Supreme Court and have them rule on it forthwith.
Then we should have impeached every President since Nixon.
Black letter law means nothing to him. Look what happen to the GM Bondholders.