(Read more: Why markets should pray for a government shutdown)
He said his message to Congress is: Do not shut down the government, do not shut down the economy.
The president said the U.S. Senate acted responsibly by voting to keep the government open and now it's up to Republicans in the House to do the same.
The president said he will not agree to delaying or defunding the Affordable Care Act, often referred to as Obamacare. Obama said health-care exchanges will be open on Tuesday even if the government shuts down.
(Read more: What you need to know about the government shutdown)
He also addressed the need for Congress to raise the debt ceiling.
"Once they vote to keep the government open, they also have to vote within the next couple of weeks...to allow the Treasury to pay the bills for the money that Congress has already spent... Raising the debt ceiling is simply authorizing the Treasury to pay for what Congress has already authorized," the president said. "Failure to meet this responsibility would be far more dangerous than a government shutdown. It would effectively be an economic shutdown."
House Speaker John Boehner responded to the president's speech, accusing Obama of grandstanding.
"The House will take action that reflects the fundamental fact that Americans don't want a government shutdown and they don't want the train wreck that is Obamacare. Grandstanding from the president, who refuses to even be a part of the process, won't bring Congress any closer to a resolution."
Obama started the press conference by saying he had just spoken with Iranian President Hassan Rouhani, the first direct communication between leaders of the two nations since 1979.
He said both he and Rouhani have directed their teams to work quickly toward an agreement on Iran's nuclear program.
"I do believe there is a basis for a resolution," on nuclear weapons, Obama said.
(Read more: Obama speaks with Iranian President Rouhani)
After the conversation, Rouhani tweeted: "In phone convo, President #Rouhani and President @BarackObama expressed their mutual political #will to rapidly solve the #nuclear issue."
In a subsequent tweet, Rouhani said, "If we can make progress on #nuclear file, other issues such as #Syria will certainly be positively affected."
Obama hasn't tweeted about their conversation. Earlier this afternoon, he tweeted: "It's time Congress puts an end to governing from one manufactured crisis to the next. Tell them "EnoughAlready."
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— Reuters contributed to this article.