From the 2002 General Instruction of the Roman Missal (GIRM) approved for the USA, which can be accessed from
romanrite.com/girm.html :
“118. … It is a praiseworthy practice to cover the chalice with a veil, which may be either the color of the day or white.”
“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.”
And: …
“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 is not clear to me why it is a “praiseworthy practice to cover the chalice with a veil”. In n. 163 and n. 183 the covering seems to be a way to indicate that the vessels are not purified. But this indication would not work if they are always covered.
Generally the use of a veil is to show special reverence for something. For example, the humeral veil.
It’s not required in the Novus Ordo but, just as the maniple, it’s use is optional. While some do use it, most, apparently, do not.