Q. Why do priests announce at Mass that non-Catholics cannot receive Communion?
A. Priests who make that announcement at Mass are simply speaking the truth and passing along what the Church teaches. Sometimes that position is not very popular and calls for a certain pastoral sensitivity, but it is the right thing to do.
There are some exceptions. The most notable is for members of the Orthodox Churches who may receive Holy Communion, as well as Penance and Anointing of the Sick, when they request them spontaneously. Outside of exceptional circumstances, clearly defined in Canon Law, Protestants are not to receive Holy Communion, and in no case are the non-baptized to be allowed to receive the Holy Eucharist.
This disposition is not a lack of friendliness with non-Catholics, but simply the consequence of what it means to receive Holy Communion. That act means the person who receives it assents to everything the Church teaches. Non-Catholics do not.
Holy Communion is also a sign of unity. Most Catholic Churches have missalettes in the pews that reprint the indications of the “Guidelines for the Reception of Holy Communion” from the United States Conference of Catholic Bishops. In part that guideline states: “Because Catholics believe that the celebration of the Eucharist is a sign of the reality of the oneness of faith, life, and worship, members of those churches with whom we are not yet fully united are ordinarily not admitted to holy Communion.”