A friend and I were discussing this. Generally speaking, it seems that it should be impossible for a person to be a kohen (rather than a chalal) and also a mamzer, because the set of women whom a kohen is forbidden to marry, and by whom his children would be chalalim, includes all of the arayos (Shulchan Aruch, Even Haezer 6:8).
However, in 7:14 it says that if a kohen impregnates such a woman the first time they have relations, then indeed the child is a mamzer but not a chalal. Chelkas Mechokek (:24) and Beis Shmuel (:37) point out that there's no real practical difference until the next generation: if the result of such a union is a daughter, and eventually a kohen wishes to marry her, he will be warned that she is forbidden to him as a mamzeres rather than as a chalalah.
So the question: is the child indeed a kohen in that case (which would seem to be implied by the statement that he is "not a chalal")? If so, then isn't there a greater practical difference - that if the child is a boy, he'll have to keep all of the laws of a kohen, and can perform birkas kohanim?
Another data point is that when checking family lineage, "we don't need to check beyond the altar" (Rambam, Hil. Issurei Bi'ah 20:2 from Kiddushin 76a), because any kohen who served in the Beis Hamikdash will have been thoroughly vetted. But if there can be a case, as above, where a person is a mamzer but still a kohen, why wouldn't he be able to perform the avodah?