The Board of Immigration Appeals recently issued a case recently stating that persons who MAY have dual citizenship are not eligible for asylum. In that case, the foreigner was a Venezuelan and his father was a citizen of Spain. The applicant was not a citizen of Spain. Because the government showed proof that the asylum applicant is eligible to become a citizen of Spain himself (and he had no fear of harm there), however, his asylum was denied.
This is a drastic change from the way the law was formerly interpreted. Blandon Law has won asylum cases for dual citizens (Venezuelan and Italian) in the past, but that seems more difficult now.