If the debt ceiling is not raised at some point the US government will be unable to meet all of its obligations.
I assume that they will make their interest payments and bond redemptions on schedule and the shortfall will be in paying social secutiry, medicare, military and other obligations. This will naturally impact aggregrate demand and generate a significant negative impact on the economy. Given the severe weakness in the economy this shock most likely would tilt the economy into a recession.
This is rather straight forward analysis, but the more severe situation would be the consequences of the government failing to redeem T bonds and/or T bills or failing to make an interest payment of these debt obligations.
Large business and financial institutions do not leave large sums sitting around not earning interest. For the most part firms invest idle balances in T bills. This reached the point long ago where banks introduced sweep accounts where they will go through a firms deposits late in the day and sweep their balance out and invest them in T bills overnight. This is where the risk free instrument comes to play a major role in the financial system and the economy. In many ways the risk free investment of T bills are like the oil in an engine. It provides the buffer or lubrication in the financial system that allow the various moving parts of the economy to move freely and not rub against each other. If the risk free instrument of the T bill is removed from the system there is nothing around of sufficient size to provide the lubrication that the system requires. Thus, if firms no longer have T bills or risk free instruments to invest in there is a danger that the financial system will seize up like an engine without oil. It becomes a question of confidence and we could quickly have a repeat of something like what happened in 2008 after Lehman Brothers went bankrupt and lenders pulled in their horns and refused to lend to otherwise good credits. This is why those claiming that the US defaulting on its debts would not have severe and wide-ranging consequences are completely wrong. It is why some of the largest financial institutions are already starting to take measures to protect themselves against this possibility.