I understand that the National Council of Young Israel has a rule that women and converts may not be elected president of a Young Israel synagogue. I think I saw a gemara -- or a comment to a gemara -- in Shabbos that said as much. What are the sources, pro and con, for the position that converts should not be shul presidents?
The Young Israel's position is based on a pasuk (verse) in Deuteronomy 17:15, as interpreted in the Gemara, and codified as law by Rambam (Maimonedes). I will provide these sources and the opposing position.
Devarim (Deuteronomy) 17:15 states: "Then you shall appoint a king over you, whom the Lord your G-d will choose: one from among your brethren shall you set as king over you, but you shall not set over you a stranger who is not your brother." Notice that the verse has some redundencies. If the rule is only that we cannot appoint a gentile king to rule over us, then the first highlighted phrase should be enough. Sifri, Shoftim, ¶ 157. The Midrash HaGadol brings down that the second highlighted phrase comes to exclude a convert.
The third redundant statement appears (although I haven't found a source exactly on point) to enlarge the halacha from not just from excluding gentiles and converts from being kings, but also from being communal leaders. Whether the rabbis learned it from the third redundancy or not, Kiddushin 76b states: "We have learned: 'Then you shall appoint a king over you from among your brethren,' all appointments of authority that you make should not be [made] except from among your brethren."
Rambam in Hilchos Melachim 1:4 similarly states: "We do not appoint a king from amongst the converts, even after several generations, until his mother is [one born] Jewish, as it is written, 'You will not set over you a stranger who is not your brother.' Not only for kingship, but also for any position of authority in Israel, neither a general nor chief over fifty people, nor chief over ten people, nor even a person appointed to verify that the water is distributed to the fields. It is superfluous to talk about a judge or a nasi, who may not be other than [one born] a Jew, as is written, 'one from among your brethren shall you set as king over you'--all the people whom you give positions of authority shall not be from other than your brethren."
With respect to whether a convert can sit on a beis din to judge his fellow Jews, there is disagreement. Rashi, in his commentary to Shabbos 102a, says a convert is allowed to judge a Jew on property matters, but not concerning capital laws (see also on Kiddushin 76b, s.v. kol mesimos.) However, numerous commentators caused Rambam to conclude, in Hilchos Sanhendrin 2:9 that "[a] Beit Din of three [judges], one of them being a convert, is disqualified until his mother is [one born] Jewish." C.f. Rif at the end of chapter 4 of Sanhedrin; Tosaphot to Yevamos 45b s.v. keivan and to Sanhedrin 36b s.v. chada; Nimukei Yosef at the beginning of chapter 12 of Yevamos; and the Ran on the Rif, end of chapter 4 of Sanhedrin. A convert can judge a fellow convert, however. E.g. Yevamos 102; Shulchan Aruch Yoreh Deah Hilchot Gerim Siman 269 si'if 11.
The key word in the analysis is "position of authority" or in Hebrew "serara." On this point, some argue that this is a weakness in Young Israel's position. Some cite Rav Moshe Feinstein's position that a convert can be a rosh yeshiva or a mashgiach ruchani of a yeshiva. In Iggros Moshe 4:26, he states "... it seems, that these are not considered authority positions in our times, that it triggers a situation of sererah, since in essence the purpose of a yeshiva is to teach to the students that which they want. And that which there is power to the leaders and the Roshei Yeshiva over the students to throw them out or to not receive them originally and similar, this is only like the authority of the master of a house over his workers, and there is not in this any matter of appointing to authority [minui l'serarah] at all...." Rav Feinstein's position in this case, I think, is not directly applicable to that of a shul president. In other cases where he discussed "serara," with respect to a (non-convert) rabbi of a shul and the head of a kashrut-certifying organization, he held that the rabbi position is a position of honor and not one of control, whereas the kashrut position was one of authority since he could overrule the business owner. Compare Id. and Iggros Moshe, Choshen Mishpat, 2:34.
Where Young Israel's position is weakest is the introduction of two factors. One is the halacha to love the convert (Deut. 10:19) which, Rav Moshe notes "obligates us to draw them close and to be lenient in all these matters." Second is a point raised in the Tzitz Eliezer 19:47 to the extent that, if the litigants concented to have a convert as a judge, then the prohibition is waived. If that is true, then community acceptance of converts, or particular converts, could also waive the prohibition against converts being president, a position that the Tzitz Eliezer adopts in the next ruling, 19:48, in the case of a convert becoming a shul gabbai.
I've been a shul gabbai. It is a position with neither honor nor power. One may argue that this is even more true of a shul president. Perhaps. But since first learning the halacha that a convert can't be a leader of a group of Jews, I have been quite machmir in applying that halacha to the position of shul presidency. So far, it has (bli ayin hara) kept me from being elected to that position, and I don't want to see a reversal in the Young Israel position (I go to a YI shul).
I can simply tell you that it is commonly done ... though there is discussion about it ... just as RIETS admits converts to its rabbinical school. (Ask me who's more likely to "rule with an iron fist", the rabbi or synagogue president ... )
The Gemara in Sanhedrin implies a Ger shouldn't stand in judgement over a native-born Jew (which doesn't sit well with me, but what do I know) ... similarly, if we derive guidelines for all positions of "dominating authority" (serara) from the Torah's laws for monarchs (as Rambam seems to do; while it doesn't appear in the Talmud it is in the Sifrei, Rambam isn't making it up himself based on his antique-Islamic-era sensibilities, as one blogger suggested), then serara would be limited to male, native-born Jews. To what extent we follow this position today, and what positions are called "dominating authority", are big questions. I recall Rabbi JD Bleich had a piece years ago in Tradition about female synagogue presidents, where I think he discussed [male] converts as well.
Rabbi Aaron Rakeffet-Rothkoff reports an oral discussion with his mentor, Rabbi Joseph Dov Soloveichik, in which Rabbi Soloveichik was uncomfortable with having female synagogue presidents out of modesty concerns; he felt the "dominating authority" argument was stretched quite thin in this context as the synagogue president gets stepped on by everyone! (But in a recent discussion on female presidents, Rabbi Mordechai Willig observed that the president's power can vary drastically between congregations.) If so I presume a male convert would pose no problem. (Well everyone who screams at and walks all over the shul president would be getting an extra sin of offending a convert, but what can we do ...)