The Treasury's Office of Foreign Asset Control had seen its manpower cut from 175 full-time staff to 11, White House spokesman Jay Carney said.
"The office is unable to sustain its core functions," he said.
The office issues new sanctions, and targets firms or individuals that violate US sanctions to punish Iran's nuclear program.
It also maintains US sanctions against Syria, terrorist organizations and proliferators of weapons of mass destruction.
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The White House and rival Republicans have been highlighting the impact of furloughs and funding cuts to make political points as the shutdown goes into a fourth day.
"(This) illustrates the consequences if the Republican shutdown continues to have on the government's missions and workers across the country," Carney said.
The Republican chairman of the House Foreign Affairs committee, Ed Royce, called on Treasury Secretary Jacob Lew to declare the work of the office essential, a status which would allow staff to return to work.
Royce argued that the workers at the office were fulfilling an emergency function critical to national security and should therefore be legally permitted to remain at their desks during the shutdown.
"The Iran sanctions regime has been developed in response to a 'national emergency' as determined by the President…I could not agree more that Iran's drive toward a nuclear weapon is a grave threat to our country," he said to Lew in a letter.
"I respectfully ask that you reconsider these ill-advised staffing decisions that undermine support for vigorous Iran sanctions and other critical national security efforts."