This is a very important point:
Failure to raise the debt will force the president to break a law — the only question is which one.
The Constitution requires the president to spend what Congress has instructed him to spend, to raise only those taxes Congress has authorized him to impose and to borrow no more than Congress authorizes.
If President Obama spends what the law orders him to spend and collects the taxes Congress has authorized him to collect, then he must borrow more than Congress has authorized him to borrow. If the debt ceiling is not raised, he will have to violate one of these constitutional imperatives. Which should he choose?
I also agree with Aaron’s answer, which is that ignoring the debt ceiling is no legally worse than the other alternatives and would have by far the least bad consequences, so that’s what Obama should do.