I noted this possibility earlier, but it now seems to be actually happening. Michael Stern (Point of Order) comments here: U.S. House of Representatives v. Obama: The Problem of Standing.
Jack Goldsmith has these thoughts: Suing the President for Executive Overreach.
And Andy McCarthy has even harsher thoughts: Boehner Is Bringing a Whistle to a Gunfight and Boehner Issues Memo Explaining His Feckless Plan to Sue Obama.
As I've discussed before, I'm skeptical of the originalist foundations of modern standing doctrine. But I'm even more skeptical that the House vs. Obama case is a "Case or Controversy" under the Constitution's original meaning. It seems to me that Speaker Boehner's principal complaint is the the President isn't doing his job right, in a way that is injuring the U.S. as a whole. Whatever one thinks "Case or Controversy" is, this can't be it -- or everyone would have standing to challenge everything.
The counter is that the House, as the body that enacted the laws, is especially harmed by the laws' non-enforcement. But this claim seems to misunderstand separation of powers. Congress has power over legislation; the President has power over execution. If the President misues his power of execution, that doesn't harm Congress (in its legislative capacity) any more than it harms everyone else -- we all have an interest in the President acting constitutionally within his domain. Similarly, one could say that we all have an interest in the Supreme Court judging a statutory case correctly; but Congress does not have a greater interest than any of the rest of us just because it passed the statute in the first place.
In contrast, in the census litigation Michael Stern discusses in his post, I agree that Congress had a distinct injury. As he explains:
The essence of the House’s complaint [in the census litigation] was an informational injury, i.e., the illegal census design would deprive it of specific information that it needed to perform its constitutionally mandated duty of re-apportioning the House of Representatives. This is quite different than the “wholly abstract” injury involved in Raines [v. Byrd, the previous cognressional standing case]. ...
By law the president is required to report the results of the decennial census to Congress in the first week of the regular session of the following year (see 2 U.S.C. § 2a (a)). If the president failed to provide the legally required census data, not only would Congress be deprived of specific information to which it was entitled, but the composition of the House itself could be affected.
Thus one can imagine ways in which the Congress' constitutional role gives it a distinct interest in certain types of executive action. But (as Stern says) this observation suggests a lack of standing in the House v. Obama litigation, because in that case there is nothing like this sort of particularized injury.
I can accept, as an orginal matter, the general proposition that the "Case or Controversy" language means at minimum that everyone can't sue everyone for everything. How modern standing law derives its particular intricacies from this basic proposition is a mystery to me. But that doesn't matter in this case, which seems fundamentally about an abstract injury common to everyone.
For some contrary views, see the links here. At best, one can say that the Constitution's original meaning in this area is poorly understood.