The Federal Government will run out of cash on around October 17 unless congress extends the Treasury Department's authority to borrow money. If it doesn't, we'll have a legal paradox—there are bills the Treasury is legally required to pay, and there are legal restrictions on the Treasury's ability to get the money to pay them.
One idea that often comes up in this context is that the government could do what a household or a firm experiencing a cash crunch would do—prioritize payments. You can let your utility bills slide for a month or two or three without actually losing service, and you can hope to scrounge up some money during the interim. The problem is that the federal government is literally not set up to do this. Jack Lew doesn't sit around in the evenings logged in to some online bill pay system deciding which checks to send and which not to send. As RBC Capital markets via Cardiff Garcia writes, the system just either runs or it doesn't:
The Treasury’s systems do not clearly mark what scheduled payments are for what reasons, so it is impractical to try to prioritize payments. And clearing systems like Fedwire do not allow defaulted securities to flow, so the system would seize. In order for the clearing systems to work, the Treasury would need to notify the market of a default almost a day before the default happened (to give everyone time to modify payments), and that is not going to happen because the Treasury will not want to declare default while Congress still has time to pass a bill.
Also the Fed does not take defaulted securities as collateral at the discount window, even if those securities are still trading at par.
My view is that in practice the responsible course of action for the executive branch and the Federal Reserve would be to just ignore the law and insist that the debt ceiling doesn't bind (or something) and try to create a constitutional crisis to forestall a financial one. But they in practice can't juggle payments. What's more, they have no more legal authority to prioritize payments than they do to borrow extra money. All administrations find themselves charged with administering programs that they don't support. President Obama can't just refuse to pay out farm subsidies any more than President Mike Lee will be allowed to refuse to pay Social Security benefits.