In Shulchan Aruch Orach Chaim 566:5, the Mechaber writes that one who is not fasting is not allowed to be a Shalicah Tzibbur. The Mishnah Berurah (Seif Katan 18) writes that if one were to find himself davening for the amud, he should say עננו in Shma Koleinu and say
"ביום צום התענית הזה". If there is no one else to daven for the amud, it is better that one is not fasting should be Shaliach Tzibbur that to miss out on Kaddish, Kedusha, and Barchu.
In 566:6, the Mechaber writes that one who is not fasting, even if he is the only Kohen in the shul, should not get called up for an aliyah. The Mishnah Berurah (Seif Katan 19) quotes the Magen Avraham who differentiates between Shacharis and Mincha. By Shacharis, the Kohen can receive the aliyah לכתחילה, whereas by Mincha, he should not get the aliyah. Some Achronim argue, but all agree that if called up (בדיעבד) the Kohen can receive the aliyah.
The Mishnah Berurah (Seif Katan 21) writes that, לכתחילה, the Kohen should leave the room at that time so that people won't say when he doesn't get called up that he is not a proper Kohen. If there is another Kohen, he doesn't have to leave, but he should tell the Gabbai not to call him up.
If the Kohen did get called up by Mincha, there is a machlokes amongst the Poskim if he should go up (because of כבוד הציבור) or not (because it may be a ברכה לבטלה). Therefore, the Kohen should ake sure not to be in the shul at that time.
If he is a תלמיד חכם and, for whatever reason, he is not fasting, he can rely on those that are lenient to avoid a חילול השם.