Why do some Christians (Coptic orthodox and catholic) kiss the priest's hand?

There are many teachings about the Holy Kiss.

Romans 16:16 NKJV

Greet one another with a holy kiss. The churches of Christ greet you.

2 Corinthians 13:12 NKJV

Greet one another with a holy kiss.

1 Thessalonians 5:26 NKJV

Greet all the brethren with a holy kiss.

1 Peter 5:14 NKJV

Greet one another with a kiss of love. Peace to you all who are in Christ Jesus. Amen.

If it is to kiss a hand or a ring or a sleeve or the air, I do not know. The tradition in the Catholic Church is the most likely example of the holy kiss. And if that example also flows out to kings of the past after the time of Christ, then that should be considered more evidence of its form. Here is a Reference you may enjoy.

In Western Catholicism it is usually connected to the anointing of a newly ordained priest's hands with the Sacred Chrism. His hands have been set aside for God to do the Lord's work, and so some people (particularly in more traditional circles) kiss the hand. This is especially common with newly ordained priests, but also happens with priests in general sometimes.

The kissing of the priest's hand in the Coptic Church is for the reasons that

1. He is a chosen vessel by God and receives a special grace given by the priesthood. He is the one who administers the Holy Eucharist to the people. His hands touch the Body and Blood of our Lord Jesus Christ, therefore, his hands are made holy.
2. It is a sign of respect and reverence to the priest who is considered a father figure.

Father Peter Farrignton of the Coptic Orthodox Church says

You do not have to kiss the priest's hand. If, as appears to be the case, you do not want to, then don't do so.

Those who do want to, will do so.

The priest handles the holy things on the altar, and especially the Body and Blood of our Lord, therefore His hands are holy. When the faithful kiss his hand they are taking a blessing. Yes, the holy elements are placed into the mouths of the believers, but that does not mean that the priest does not have a special grace, as has already been explained.

I have Roman Catholic Christians worship in my own Churh sometimes and many of them kiss my hand to receive a blessing, or my cross, or both. None of those who worship with me shake hands with me during the liturgy or when seeking a blessing. Though these are only a few folk and may not be representative of those Roman Catholics you know.

If you object to kissing a priest's hand then don't do so.

I have to say that I never find it a source of pride that someone kisses my hand seeking a blessing. I am well aware of my faults and weaknesses. If they receive a blessing it is from God in any case, using my unworthiness, and not from me.

Father Peter

Source

Furthermore, His Grace Bishop Youssef, Bishop of the Coptic Southern Diocese of the United States says

We kiss the hands of bishops and priests because in the context of the Divine Liturgy, the hands hold and distribute the holy Body and Blood of our Lord, God and Savior Jesus Christ! This is an enduring and centuries old tradition in the Eastern Christendom that has a purely metaphorical connotation; of course, this is not to leave out an underpinning element of paternal respect and affection as well. Kissing of the hands is also done amongst clergy. The blessing of the priest has a marvelous efficacy as being an exercise of the mysterious power with which he is entrusted. Through the prayers offered by the priest, the Holy spirit changes bread and wine into the Body and Blood of Christ. The priest bestows special sanctity upon Christians and upon the objects. The hand of a priest is, therefore, an instrument for imparting Divine Grace. For this reason Orthodox Christians throughout the centuries customarily kiss the hand of their priest.

Source