Ayatollah Ahmad Janati made the declaration in a published letter just one week before parliament is to approve a list of candidates.
Iran will simultaneously hold city and village councils elections and presidential elections on May 19.
Iran's council election law states candidates from recognised religious minorities may run as candidates in city and village elections as long as they "Believe in and demonstrate their commitment to their own religious principles in practice."
Discrimination based on religious affiliation is not new in Iran. Religious minorities, especially members of the Bahai faith, which was founded in the country, are not recognised in Iran's constitution and suffer from serious discrimination.
The Guardian Council is an influential body made up of clerics and jurists and has in recent years disqualified a significant number of candidates. However, it is Iran's parliament that is legally empowered to vet city and village council candidates - not the Guardian Council.
According to Human Rights Watch, Esfandiar Ekhtiari, a member of parliament, has called the council's letter "unconstitutional."
It won't be clear if the parliament will adhere to the Council's stance on non-Muslims not running in the local elections until the list is published next week.