A baraita on Shabbat 24a (see also Tosefta Berakot 3:10) states that one should include material relevant for fast days at Arvit, Shaharit and Minhah of fast days. This seems to be something like the עננו paragraph said even today on fast days in the Amidah. Similarly, Yerushalmi Ta'anit 2:2 reports R. Huna saying that one must mention even individual, elective fast days both at night and during the day in one's Amidah. A Geonic tradition (Otzar Hageonim Shabbat Responsa #76) rules that the inclusion of this material, while mandatory in Minhah, is optional in Arvit and Shaharit, without explanation. Rashi reports a Geonic tradition that goes further: we don't include this material in Arvit and Shaharit, for fear that the person saying it will not complete the fast and will end up having lied about it being a fast day. Only at Minhah (which, seemingly, is said towardds the end of the day) do we feel confident enough to complete the fast such that we say what we know as עננו. In the end, common practice has engaged a kind of compromise position: the leader on a fast day says עננו as a separate blessing during the repetition in both Shaharit and Minhah (there being no repetition at Arvit), while individuals only include it at Minhah.
This last tradition seems to take a clear stand that one who is not fasting certainly cannot say עננו and would thus seem to preclude someone who is not fasting from leading on a fast day.
Indeed, a Geonic tradition cited first in the name of R. Yehudai Gaon (see Otzar Hageonim Ta'anit Responsa #58-59) states that one who is not fasting should not lead on a fast day, because the person cannot honestly say עננו--"answer us"--because the person is not fasting. Tur disagrees with this ruling and says that there is no reason that a person cannot say עננו ביום תענית הזה--"on this fast day", as opposed to "our" or "my"--even if not fasting, given that it is indeed a fast day in the community. [Tur might maintain the Geonic tradition above regarding not saying עננו in Arvit and Shaharit and its implication that one not fasting should never say עננו as being about individual fast days; communal fast days have a reality to them that transcends the individual and thus even those not fasting can talk about them litrugically as fast days.] Tur ends up concluding that it is preferable to have someone who is fasting as shatz, but clearly allows being lenient, such as if there is no one else prepared to do it. We might add that one could also be lenient if feelings would be severely hurt.
In any event, Beit Yosef backs away from this and returns to the ruling of the Geonim that one who eats is disbarred from leading on a fast day, and so he rules in the Shulhan Arukh in OH 566:5.
Magen Avraham merely notes that if the non-faster already began leading, we can take a compromise position between Tur and SA: have the leader say עננו ביום צום התענית הזה in שומע תפלה (where individuals add it in Minhah), as opposed to adding it in between the seventh and eighth berakhot, where the leader normally adds it. That way, we don't add an entirely new berakhah into the repetition, in case SA really takes a hard line on this issue, but we can rely on the Tur's logic not to consider such a statement to be a lie. It does seem, however that MA would not want you to say תעניתנו, feeling that to be dishonest, but rather ביים צום התענית הזה. [Note that Hayyei Adam understood the Magen Avraham to permit adding עננו as its own berakhah in this case, outright relying on the Tur if the person already started leading.]
With regard to Yom Kippur, it is noteworthy that the only axis of concern in the debate seems to be the truthfulness of saying עננו when not fasting. The Yom Kippur liturgy features no such statements and the leader is thus never in a position of acting as if s/he is fasting even though s/he is not. Having someone who is justified in eating on Yom Kippur lead thus does not seem to present any of these issues.
Following the Shulhan Arukh, one should never plan to have someone who is not fasting lead on a fast day, and even on Yom Kippur, where the issues raised here do not apply, one should think carefully about what sort of exemplar one wants leading the community on a day when almost everyone else in the room is fasting. It seems strongly preferable to have someone as a leader who is joining the community in that experience and can thus emotionally connect with them in this regard. Even if this means that pregnant and nursing women order to eat and drink minimally by their doctors--without such an order, pregnant and nursing women have the same fasting obligation as anyone else--cannot lead during those years of childbirth, having a leader in sync with the community's practice on that day seems a strong value. But because Yom Kippur is different, perhaps there is more room to be flexible with someone who already accepted a paid position or a pregnant woman ordered to eat by her doctor who a) did not know she would be pregnant when she accepted the position or b) will be drinking in such small quantities throughout the day (which she should ideally be doing anyway, even if ordered to drink), that she will still meaningfully experience Yom Kippur as a fast day.
Certainly, if someone was already assigned the position of leading and it is impossible to replace them, either for lack of someone else qualified or for fear of shaming someone, then one can rely on the Tur to let them lead on a fast day and certainly on Yom Kippur, where arguably no one thinks there is a core problem. Otherwise, one should make every effort to only choose prayer leaders on fast days who are themselves fasting.