According to the U.S. Department of State, the Executive Order does not restrict the travel of dual nationals, so long as they hold the passport of an unrestricted country and possess a valid U.S. visa.
“Our Embassies and Consulates around the world will process visa applications and issue nonimmigrant and immigrant visas to otherwise eligible visa applicants who apply with a passport from an unrestricted country, even if they hold dual nationality from one of the seven restricted countries.
At this time, there are no changes to the Visa Waiver Program (VWP). However, as before the Executive Order, dual nationals of Iraq, Iran, Syria, and Sudan are not eligible for the Visa Waiver Program and must have a visa for travel. Such individuals may apply for a visa as needed at a U.S. Embassy or Consulate. Dual nationals of Yemen, Libya, and Somalia and a VWP country may continue to travel under the VWP unless they have travelled to any of the seven countries on or after March 2011, in which case they must first apply for a U.S. visa.
The Executive Order does not restrict the travel of dual nationals from any country, so long as they hold the passport of an unrestricted country and possess a valid U.S. visa, if required.
The U.S. Department of State has posted additional information on the affect of the Executive Order on the visa process for nonimmigrants and immigrants.
The U.S. Customs and Border Protection has posted a Frequently Asked Questions (FAQ) page on their website with further information and developments regarding entry to the U.S. under the Executive Order.