On Good Friday, the priest is the first one to venerate the cross and when he does, he will often take off his shoes.
Why is that?
First of all, taking off shoes before the veneration of the cross is written in the instructions in the Roman Missal.
For the Adoration of the Cross, first the Priest Celebrant alone approaches, with the chasuble and his shoes removed, if appropriate.
This tradition has been part of the Roman Catholic liturgy for centuries and is clearly derived from the book of Exodus, where God asked Moses to take off his shoes.
God said: Do not come near! Remove your sandals from your feet, for the place where you stand is holyground.Exodus 3:5
Venerating the cross is a holy act, and in remembrance of Jesus’ profound act of love, the priest (and sometimes even people in the congregation) will take off their shoes to show their gratitude to God.