If a father does not arrange pidyon haben for his firstborn son, what happens? Is the son by default given to service of the Kohanim? Is it the duty of the son to become a rabbi?
If the father neglects to do it the son redeems himself at his first opportunity. (Shulchan Arukh YD 305:15)
It's possible that the local rabbinic authority is responsible to arrange for the Pidyon Haben if the father doesn't do it (Be'er Heitev on YD 305:15). If not, the son should do it himself when he reaches Bar Mitzvah (Pitchei Teshuva, ad loc. "עיין בת׳ זכרון יוסף שכתב דהיינו שיהיה בן י״ג שנה דוקא.")
The way I understand it, because this obligation carries on to the child, there's no possibility of anyone being actually given over to the Cohen. Or maybe it's the other way around...