At least for Mass, at least one cloth is required.
From the 2002 General Introduction to the Roman Missal (GIRM) which can be accessed from
romanrite.com/girm.html :
“*The Articles to Be Prepared *
117. The altar is to be covered with at least one white cloth. …”
This one cloth is required at the start of Mass. During Mass a second small cloth, the corporal, is added:
“297. The celebration of the Eucharist in a sacred place is to be carried out on an altar; but outside a sacred place, it may be carried out on a suitable table, always with the use of a cloth, a corporal, a cross, and candles.”
“304. Out of reverence for the celebration of the memorial of the Lord and for the banquet in which the Body and Blood of the Lord are offered on an altar where this memorial is celebrated, there should be at least one white cloth, its shape, size, and decoration in keeping with the altar’s design. When, in the dioceses of the United States of America, other cloths are used in addition to the altar cloth, then those cloths may be of other colors possessing Christian honorific or festive significance according to longstanding local usage, provided that the uppermost cloth covering the mensa (i.e., the altar cloth itself) is always white in color.”
Perhaps the priest only intends to remove a frontal cloth. This could be done, according to a book by G. Thomas Ryan, The Sacristy Manual: “Several years before Vatican II, the Roman rite dropped the requirement of a frontal or antependium. As altars became free-standing, it became all the more obvious that mandates for decorating only the front should be discontinued.” (G. Thomas Ryan, The Sacristy Manual, published by Liturgy Training Publications, Chicago, 1993, ISBN 0929650921, page 49.)