U.S. export control regulations generally treat Lawful Permanent Residents (people who have “Green Cards”) as though they are Americans. For instance, an Iranian citizen might not be able to access certain devices or technical know-how in one of our labs until she became a Lawful Permanent Resident. The same approach would apply if she were lawfully in the U.S. under special protection, such as is offered to those who are allowed to stay in the U.S. as refugees, or those who have been granted asylum.
However, please note that the U.S. Government may take an interest in Lawful Permanent Residents who are citizens of certain countries and with whom we expect to share certain controlled items or information that the U.S. Govt. believes an individual’s home country might use to undermine our national security. It is for this reason that we will sometimes ask for an individual’s status in the U.S., as well as their current and past citizenships.