The chazzan, or sheliach tzibur, who leads the prayers is an agent of the congregation. As such, it is necessary for him to be fitting to represent the community. The Mishnah in Taanis states that on public fast days they would be very particular who would be chosen to lead the prayers and that they would only choose someone who was not a sinner, nor had the reputation of being a sinner in his youth. (Gemara Taanis 15b)
The Rishonim explain that this idea of not using a sinner as the sheliach tzibur is not specific to public fast days, but rather should be avoided on a regular weekday as well. (Sefer Hamanhig Taanis 2) As the Rif writes in a response (281), “A chazzan who fouls his mouth (ie. uses foul language) or sings non-Jewish (Islamic) songs should be removed.” The Shulchan Aruch (53:4) codifies this as normative halacha. He writes, “The leader must be one who is fit. Who is “fit”? One who is free of sin; who has never had a poor reputation, even in his youth; who is humble; who is acceptable to the nation; who is skilled at [liturgical] chanting; whose voice is pleasant; who is accustomed to reading Torah, Prophets, and Writings (i.e. biblical text).” What requires clarification is what actions are defined as sins that would prevent one from being the sheliach tzibur and what practically should one do if someone who is non-observant wishes to be chazzan and lead the congregation on the yahrtzeit of a family member.

Section 1: Details Of The Prohibition
As noted above, the poskim write that one should not appoint a sinner to serve as the sheliach tzibur. The Sefer Mishpat Tzedek (2:75) writes that one who is disqualified from leading the congregation due to his sins cannot serve as the sheliach tzibur even on a temporary basis.
Rav Ovadia Yosef zt”l (Chazon Ovadia Yomim Noraim page 36) rules that it is better to pray without a minyan than to pray with a minyan being led by a sinner. A similar ruling was expressed by the Lubavitcher Rebbe zt”l. (Shulchan Menachem Vol. 1 Page 198)

Section 2: Which Sins Invalidate Someone From Serving As Chazzan
Publicly Desecrating Shabbos and Yom Tov– As noted in previous articles the Gemara in Chullin 5a states that if one serves idolatry he is punished that he has the hallachic status of a non-Jew. Similarly, one who desecrates Shabbos in the presence of ten Jews has the halachic status of a non-Jew. The reason for this is that we keep Shabbos because Hashem rested on the seventh day after creating the world in the first six days. One who does not keep Shabbos is thereby questioning the fact that Hashem rested after creating the world. Therefore, one who desecrates Shabbos is akin to serving Idolatry and is therefore punished that he strictly treated as non-Jew in halacha. Since he has the status of a non-Jew he cannot be counted towards the minyan. Based upon this halacha, one who publicly desecrates Shabbos or Yom Tov cannot serve as the chazzan. This is the view of many poskim, including Rav Ovadia Yosef zt”l.
The Lubavitcher Rebbe zt”l writes, “Unfortunately here in this country it is necessary to make it clear that whoever desecrates Shabbos publicly cannot be the chazzan. He should not be despised or embarrassed, and on the contrary, he should be approached and influenced to come to the synagogue out of the hope that tomorrow he will repent, but even so, since at this time ‘tomorrow’ has not yet arrived, he is in a position and situation that is not yet capable of serving as chazzan, He should not be made embarrassed by this, but it can be explained to him that “there is not a righteous person in the land who does good and does not sin,” and everyone fails sometimes, etc. On the contrary, he will realize by himself of the necessity to repent and correct his actions.”
The Sefer Shaarei Deah (2:35) rules that one who desecrates the second day Yom Tov outside of Eretz Yisroel is likewise disqualified from serving as the chazzan.
One Who Intermarried– It is clear from the words of the Rama that one who has sexual relations with a non-Jew, and all the more so one who intermarried cannot serve as chazzan. (Refer to Rama ibid.)
Shaves With A Straight Blade– One who shaves with a straight blade which is biblically forbidden would disqualify him from serving as a chazzan. (Chazon Ovadia ibid.)
Public School– Rav Ovadia Yosef zt”l writes that one who chooses to send his children to public school and not to yeshiva or bais yaakov is not fit to serve as a chazzan.
Non-Believers and Heretics– The Mishnah Berurah (126:2) rules, “It is universally known that one who denies the resurection of the dead, or if he does not believe in the coming of Moshiach. And all the more so if he does not believe that the Torah is from Heaven or in the idea of reward and punishment he is an apikores and one cannot allow him to serve as chazzan.”
Rav Moshe Feinstein zt”l writes many times that Reform or Conservative Rabbis have the status of an apikores in halacha and are disqualified from serving as witnesses etc. Accordingly, they would not be allowed to serve as chazzan as well. (Refer to Igros Moshe E.H. 1:135, E.H. 4:46 and E.H. 2:17)

Section 3: Tinuk Shenishbah
We have discussed in previous articles that many of those that are not observant are classified as tinokos shenishbu, meaning their sins and lack of religious observance is not due to their rebellion against God, but rather due to their ignorance and lack of Jewish education. Many Jews are raised thinking that Observant Jewish life is extreme and that keeping Shabbos is the “crazier” act. Such Jews are considered Jews in halacha and the question then is whether they can serve as the chazzan. Does the fact that they are classified as tinok shenishbah remove the disqualification or do we look at the fact that they are practically deficient in these areas, albeit not their fault, that are necessary to lead the congregation.
The indication of Rav Ovadia Yosef zt”l is that one may still not serve as the chazzan. He writes that a tinok shenishbah who repents can serve as the chazzan, which implies that a tinok shenishbah who does not yet repent is still restricted from doing so. However, Rav Dovid Tzvi Hoffman zt”l seems to allow a Shabbos desecrator who is a tinuk shenishbah to serve as the sheliach tzibbur. (Refer to Shu”t Melamed L’Hoil 1:29)
Rav Avraham Edelstein shlit”a, in his work The Laws of Outreach, writes, “Rav Dovid Cohen shlit”a allows someone who believes in Hashem but who is not observant to lead the davening”.

Section 4: Ashrei and Onward
When adhering to the law of preventing a sinner from serving as chazzan, one should avoid embarrassing him or causing an argument. One can say that there is already a chazzan appointed etc. However, at times this is not feasable and if you refuse him serving as chazzan it will cause strife etc. In such a situation, Rav Moshe Feinstein zt”l permits him serving as chazzan for the second part of Shacharis, from Ashrei, Uva Letzion until the end of Shacharis. This may help to mitigate the argument. (Cited in Rivevos Efraim 4:44:149)
Rav Yosef Eliyah Henkin zt”l writes, “Regarding a chazzan, of course according to the strict law he needs to be kosher and proper (religiously). At the very least from Barchu until after Shemoneh Esrei one should be proper”. From the fact that he emphasized the need for the chazzan to be fit until after Shemoneh Esrei, it would seem from Ashrei onwars, which is after Shemoneh Esrei, that he too would permit, under extenuating circumstances, to allow one who is not fit to serve as chazzan to lead from Ashrei onward. (Kisvei Rav Henkin Shu”t Ibrah 2)

Summary and Conclusion
The chazzan, or sheliach tzibur, who leads the prayers is an agent of the congregation. As such, it is necessary for him to be fitting to represent the community. The poskim therefore write that one who is deemed as a sinner should not serve as the chazzan and that it is better to pray without a minyan than to pray with a minyan being led by a sinner.
The poskim discuss which sin would disqualify someone from serving as chazzan. The sin of publicly desecrating Shabbos and Yom Tov, intermarriage, using a blade to shave, sending one’s kid to public school are some of the sins that prohibit you from serving as chazzan.
If one is raised without a Jewish education they are classified as a tinok shenishbah. The poskim discuss whether the fact that they are classified as tinok shenishbah remove the disqualification or do we look at the fact that they are practically deficient in these areas, albeit not their fault, that are necessary to lead the congregation. The conclusion of Rav Dovid Tzvi Hoffman zt”l is that a tinok shenishbah may serve as chazzan. Similarly, Rav Dovid Cohen shlit”a allows someone who believes in Hashem but who is not observant to lead the davening.
When adhering to the law of preventing a sinner from serving as chazzan, one should avoid embarrassing him or causing an argument. One can say that there is already a chazzan appointed etc. However, at times this is not feasable and if you refuse him serving as chazzan it will cause strife etc. In such a situation, Rav Moshe Feinstein zt”l permits him serving as chazzan for the second part of Shacharis, from Ashrei, Uva Letzion until the end of Shacharis. This may help to mitigate the argument.