I didn’t know he couldn’t touch the monstrance. I’ve seen the priest/deacon touch the monstrance many times. He always uses the veil when he lifts it up, but I don’t know that he isn’t allowed to touch it at all during adoration.
I heard on EWTN that the reason the priest uses the veil during the Benediction (the blessing) with the Blessed Sacrament is to show that it is Jesus blessing us, not the priest. Thus the priest hides his own hands in order that we focus on Jesus in the Holy Eucharist as He blesses us.
Additionally the priest usually/always touces the monstrance when placing the Blessed Sacrament and reposing. As for the reason for the veil, Didi has what I was going to say.
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