U.S. export control regulations generally treat Lawful Permanent Residents (people who have “Green Cards”) as though they are American persons. For instance, an Iranian citizen might not be able to access certain devices or technical know-how in one of our labs until she becomes a Lawful Permanent Resident of the United States. 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.
Please note that there are instances in which the U.S. Government may take an interest in a Lawful Permanent Resident’s prior citizenships, when more highly restricted goods and information are involved.