In capsule, these are the guidelines for the reception of Communion by non-Catholic Christians in any Catholic Church (Roman or Byzantine):
Because Catholics believe that the celebration of the Eucharist is a sign of the reality of oneness of faith, life, and worship, members of those churches not yet fully united with the Catholic Church are ordinarily not admitted to Holy Communion.
Members of the Orthodox Churches, the Assyrian Church of the East, and the Polish National Catholic Church are allowed to receive communion in any Catholic Church,without the prior permission from the diocesan bishop. However, they are urged to respect the discipline of their own churches.
For Christians (like Anglicans, Lutherans, and other Protestant denominations) other than the above, prior permission according to the directives of the diocesan bishop and the provisions of Canon Law is necessary before communion is granted.
This was the case for Prime Minister Tony Blair (Anglican) of the UK when he was allowed to receive Holy Communion from the Holy Father when he was invited to attend a private Mass offered by the Pope during his visit to the Vatican last Feb. 22nd. The dispensation was granted by the Cardinal Prefect of the Secretariat of State. (BTW, Tony Blair's wife and all their children are Catholic.)