If a Cohen couple gets an embryo donation and their first born is a boy, do they have to do pidyon or is the baby a Cohen?
What do you mean by "Cohen couple"?
If dad is a Cohen but mom is not, then we'd have to argue that this child's halachic father is "dad." As "dad" is not the biological father, I don't see how that argument could hold. (As I understand it, biological fatherhood is a necessary but not sufficient condition for halachic fatherhood.)
If mom is a Cohen, we get to the question of halachic motherhood -- is it defined as egg mother or host mother? Today a lot of rabbis will throw up their hands and say "it's left in doubt!" However, many rabbis in Israel, as well as the rabbinic leadership of Baltimore, say it's defined by the egg. The previous generation of American poskim, including Rabbi Aaron Soloveichik, felt it's defined by the birth mother.