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WASHINGTON — When President Obama announced last month that he was barring a Baghdad bank from any dealings with the American banking system, it was a rare acknowledgment of a delicate problem facing the administration in a country that American troops just left: for months, Iraq has been helping Iran skirt economic sanctions imposed on Tehran because of its nuclear program. The little-known bank singled out by the United States, the Elaf Islamic Bank, is only part of a network of financial institutions and oil-smuggling operations that, according to current and former American and Iraqi government officials and experts on the Iraqi banking sector, has provided Iran with a crucial flow of dollars at a time when sanctions are squeezing its economy.