Can the Sunday responsorial Psalms be changed at the discretion of the cantor or priest?

No one has the authority to make unapproved changes in the liturgy. Nor can a hymn be substituted for a responsorial Psalm. The General Instruction of the Roman Missal clearly states that “Songs or hymns may not be used in place of the responsorial Psalm” (GIRM 61).

The Code of Canon Law says:

In celebrating the sacraments the liturgical books approved by competent authority are to be observed faithfully; accordingly, no one is to add, omit, or alter anything in them on one’s own authority. The minister is to celebrate the sacraments according to the minister’s own rite. (CIC 846 §1-2)

According to the GIRM:

After the first reading comes the responsorial Psalm, which is an integral part of the Liturgy of the Word and holds great liturgical and pastoral importance, because it fosters meditation on the word of God. The responsorial Psalm should correspond to each reading and should, as a rule, be taken from the Lectionary. It is preferable that the responsorial Psalm be sung, at least as far as the people’s response is concerned. Hence, the psalmist, or the cantor of the Psalm, sings the verses of the Psalm from the ambo or another suitable place. The entire congregation remains seated and listens but, as a rule, takes part by singing the response, except when the Psalm is sung straight through without a response. In order, however, that the people may be able to sing the Psalm response more readily, texts of some responses and Psalms have been chosen for the various seasons of the year or for the various categories of Saints. These may be used in place of the text corresponding to the reading whenever the Psalm is sung. If the Psalm cannot be sung, then it should be recited in such a way that it is particularly suited to fostering meditation on the word of God. In the dioceses of the United States of America, the following may also be sung in place of the Psalm assigned in the Lectionary for Mass: either the proper or seasonal antiphon and Psalm from the Lectionary, as found either in the Roman Gradual or Simple Gradual or in another musical setting; or an antiphon and Psalm from another collection of the psalms and antiphons, including psalms arranged in metrical form, providing that they have been approved by the United States Conference of Catholic Bishops or the diocesan bishop. Songs or hymns may not be used in place of the responsorial Psalm. (61)