He can't shirk the responsibility completely, but he can hire someone else, in this case a mohel, to serve as his agent to accomplish the mitzvah on his behalf. Which most fathers opt to do. [here's a story of a father who opted to do the bris]
What is the father's role at the bris?
- He should show up.
- He must appoint the mohel as his shaliach/ agent to fulfill the mitzvah on his behalf.
- He says the blessing on bringing his son into the covenant - להכניסו בבריתו של אברהם אבינו - l'hakhneeso bivreeso shel avraham avinu.
- He usually produces his son's name for the person who announces the baby's Hebrew name.
Many fathers speak after the ceremony. Either before or during the celebratory meal.
It is always a good idea to be supportive of the baby's mother, who is often (though certainly not always) going through a rough ordeal with the bris and all.
Be there for her, and try to leave the sarcastic remarks at the door. Unless of course you can't control yourself. But she probably already knows that about you. ;)