From a seminarian reader:
I have a question regarding serving the extraordinary form of the Mass which I was hoping you would be so kind as to answer for me.
Toward the end of the Mass, after communion, the missal has to be moved from the Gospel side back to the Epistle side; I have seen where the servers both come up, with one taking the missal and moving it thus, while the other takes the chalice veil and moves it to the priest’s left. But the vice-rector of my seminary has told me that this is not as he remembers it from his youth. Should the priest be the only one to touch the chalice veil, or is it permissible to have the servers move it in this limited circumstance?
Thank you in advance for your consideration and elucidation.
Servers may touch the chalice veil.
They can certainly touch the sacred vestments, for the priest and the chalice. As a matter of fact, the servers can and should help Father vest before and divest after Mass. Serves also touch the chasuble during the elevation. They deal with the biretta was well and hold copes and humeral veils when necessary.
In some places servers are trained to take the chalice veil over to the Gospel side. When two serve, one takes the veil and the other the book stand and they "swap" positions, as it were.
This is mostly a practical matter, especially when the altar may not be very wide: you get the chalice veil out of the way for the arrival of the bookstand.
Sometimes the server, especially when serving alone, would just place the veil on the mensa of the altar and go about his business. When two serve, sometimes the server with the veil holds it out for the priest so that he can take it and drape it over the chalice. Thus enters the pesky problem of the server, doing the celebrant a good turn by bringing the veil to the Gospel side, handing the priest the veil the wrong way. 
I don’t remember ever seeing anything written that obliges the removal of the chalice veil to the Gospel side after Communion. Frankly, I am used to having it simply remain on the Epistle side. Since I tidy up fairly quickly after the ablutions, it isn’t a problem.
But… in short… yes, the server can handle the veil. And yes… it is very common for the server to shift it to the Gospel side after Communion.
Let us not forget that the GIRM says a chalice veil really should be used in the Novus Ordo! At least a white veil.
I think a bare chalice for the beginning and end of Mass is disrespectful. It can be even ugly thing to have to see.
A chalice has its dignity!
The unveiling of a chalice and its veiling help to define different moments of Holy Mass.
Vesting the chalice in a matching veil, underscores the unity between the priest and the gifts on the altar. Christ is priest/victim. The priest is also victim during Mass.
There is also something of the nuptial symbolism of the unveiling of the chalice during Mass, during this sacred action wherein we encounter mystery, and hidden mystery which is in part revealed during Mass.
