I think a deacon should remove the veil if present. If not, an instituted acolyte should remove the veil, if present. If neither of these an altar server should do it.
From the 2002 General Instruction of the Roman Missal (GIRM) approved for the USA, which can be accessed from
romanrite.com/girm.html :
For “Mass Without a Deacon”: “139. When the Prayer of the Faithful is completed, all sit, and the Offertory chant begins (cf. above, no. 74).
An acolyte or other lay minister arranges the corporal, the purificator, the chalice, the pall, and the Missal upon the altar.”
Under the heading “The Duties of the Acolyte” it has: “190. If no deacon is present, after the Prayer of the Faithful is concluded and while the priest remains at the chair, the acolyte places the corporal, the purificator, the chalice, the pall, and the Missal on the altar.”
For “Mass With a Deacon”: “178. After the Prayer of the Faithful, while the priest remains at the chair, the deacon prepares the altar, assisted by the acolyte, but it is the deacon’s place to take care of the sacred vessels himself.”
There is no mention of the veil being put on the altar. From the 2002 GIRM: “306. Only what is required for the celebration of the Mass may be placed on the mensa of the altar: namely, from the beginning of the celebration until the proclamation of the Gospel, the Book of the Gospels; then from the Presentation of the Gifts until the purification of the vessels, the chalice with the paten, a ciborium if necessary, and, finally, the corporal, the purificator, the pall, and the Missal.”
So I would expect the veil to removed before the chalice reaches the altar.
The word “veil” only appears once in the 2002 GIRM:
“118. The following are also to be prepared:
a. Next to the priest’s chair: the Missal and, as needed, a hymnal;
b. At the ambo: the Lectionary;
c. On the credence table: the chalice, a corporal, a purificator, and, if appropriate, the pall; the paten and, if needed, ciboria; bread for the Communion of the priest who presides, the deacon, the ministers, and the people; cruets containing the wine and the water, unless all of these are presented by the faithful in procession at the Offertory; the vessel of water to be blessed, if the asperges occurs; the Communion-plate for the Communion of the faithful; and whatever is needed for the washing of hands.
It is a praiseworthy practice to cover the chalice with a
veil, which may be either the color of the day or white.”
For “Mass at Which Only One Minister Participates” the 2002 GIRM has:
“255. Before Mass, the necessary vessels are prepared either at the credence table or on the righthand side of the altar.”
The other use of the veil is so that the chalice is suitably covered because it has not been purified during Mass. This is referred to in 2002 GIRM 163 and 183:
“163. … If the vessels are purified at the altar, they are carried to the credence table by a minister. Nevertheless, it is also permitted, especially if there are several vessels to be purified, to leave them
suitably covered on a corporal, either at the altar or at the credence table, and to purify them immediately after Mass following the dismissal of the people.”
“183. When the distribution of Communion is completed, the deacon returns to the altar with the priest and collects the fragments, if any remain, and then carries the chalice and other sacred vessels to the credence table, where he purifies them and arranges them in the usual way while the priest returns to the chair. It is also permissible to leave the vessels that need to be purified,
suitably covered, at the credence table on a corporal and to purify them immediately after Mass following the dismissal of the people.”
It seems that covering the chalice indicates that it has not been purified. So I think it would be better not to cover it at the end of Mass if it had been purified. But perhaps there could be different types of covers.
