Recently, we had a situation where a kohein who is married to a goy was called up for an aliya. The policy in the shul is to give aliyos to ovrei aveiros (which will hopefully change one day), but the question is whether a kohein is worse. The Shulchan Aruch (128:40) writes that a kohein married to a gerusha is not called for the first aliya and looses all privileges of being a kohein, but implies that he is allowed to get an aliya. I assume that the same would hold true with a kohein married to a goy, even though it is obviously a much more sever issur.
R' Moshe (O.C. 2:33) has an interesting teshuva to the Sridei Eish, regarding a kohen who is married to a gerusha. R' Moshe says he agrees with the sridei eish that the kohen is not allowed to get the first aliya but is allowed to get an aliya. He also comes to answer the question of the pri megadim: Since a Talmid Chacham comes before a Kohein, because kedushas hatorah takes precedence, certainly a Yisroel who keeps mitzvos should come before a kohein who does aveiros. Why is the halacha that only aveiros of kehuna make the kohein loose his rights to get the first aliya, but for other aveiros he will not loose his rights?
R' Moshe elaborates in explaining that a talmid chacham doesn't have more kedusha then a kohein, he merely has rights of kedima because others have a higher responsibility to honor him. Kedusha comes only from being chayev in more mitzvos. Therefore, a talmid chacham comes before a kohein due to the responsibility to be mechabeid him, but his kedusha is still less. Therefore, a kohein who is a rasha would come before a yisroel who is shomer mitzvos, since there isn't any chiyuv to honor a yisroel who is shomer mitzvos and the kohein is obligated in more mitzvos still has more kedusha.