John Yoo has a short, thoughtful treatise about the contest sure to develop between Congress and the White House with a White House seeking to conceal as much information as possible. See The Coming Clash on Benghazi. He predicts there will be three major legal issues that will need to be resolved: executive privilege, the right against self-incrimination, and contempt.
Here is a summary:
Executive Privilege - Congress has the power of oversight and the right to get the information necessary to do its job. However, the President's people might claim "Executive Privilege" to protect information that should stay secret to protect diplomatic or military operations. But there's a snag. To make that claim would be an admission that the President actually received information and advice during the attack. Furthermore, the damage at Benghazi has already been done, and surely the diplomatic and military practices in place there are not still in use elsewhere, so any disclosure wouldn't be damaging. Besides, the committee could agree to keep it secret anyway.
Taking the 5th - It's likely that the President's people may take the 5th. (The one amendment the President likes, as the old joke goes.) The committee could offer immunity, but if they don't or the witnesses still won't talk then the committee could find them in contempt.
Contempt - Now we have a problem. The Executive Branch is in charge of criminal prosecution, so a contempt charge would be subject to an Attorney General who is extremely dedicated to the President. So expect no attempts at prosecution. That leaves Congress to exercise its own power to punish for contempt. John Boehner has already expressed a reluctance to do that. So it's anybody's guess whether Congress would do anything. To quote Yoo's closing sentence:
In the end, Congress has the authority to respond by invoking its own constitutional powers, but whether the House will flex its constitutional muscles will require political courage that is rare in Washington.
On the other hand, a lot of people will be watching. And it will be much easier to be courageous when millions of eyes are on them.