There is rain in the forecast for this weekend in NY, so it pays to review the halachos of yeshivas sukkah on the first nights beforehand to be prepared for contingencies. If it is raining enough to spoil your meal in the sukkah, under normal circumstances you should leave the sukkah and eat indoors. Remaining in the sukkah under those circumstances is not a “chumra” of any kind; it is an act of foolishness. The first night of sukkos, however, may be different. The gemara derives from a gezeirah shava to Pesach that just like there is a mitzvah to eat matzah on the first night of Pesach, there is a mitzvah to eat a k’zayis in the sukkah on the first night of Yom Tov. This is why we wait until dark before eating in the sukkah – just like the Pesach seder cannot be done before dark, this mitzvah of eating in sukkah cannot be done before dark. The Mishna Berurah has an interesting chiddush based on an extension of this idea: just as (according to the GR”A) there is a kiyum mitzvah in eating matzah all 8 days of Pesach, so too there is a kiyum mitzvah in eating in the sukkah all 8 days of the chag.
The Rishonim debate whether this mitzvah of eating a k’zayis on the first night applies even if it rains. There are two possible ways to explain the debate: 1) Is the chiyuv to eat learned from the gezeirah shavah an independent din with its own parameters or an extension of the chiyuv of mitzvas sukkah? If it is an extension of the chiyuv of sukkah, then the same exemption for rain that ordinarily applies to sukkah applies in this case as well; if it is a new independent chiyuv, then the exemption from the mitzvah of sukkah in case of rain has no bearing. 2) A second possible explanation: Does rain exempt the chovas hagavra of the person to eat in sukkah, or does it disqualify the cheftza of the sukkah? If rain disqualifies the sukkah, there is no point to eating therein. If rain exempts the individual from the normal mitzvah of sukkah, there still remains on the first night the additional obligation of eating k’zayis generated by the gezeirah shava. The Rama (639:5) paskens like the view of the Ra"n that there is a chiyuv to eat in sukkah even if it rains; the Vilna Gaon paskens like the view of the Rambam that there is no chiyuv. Since we are faced with a machlokes regarding a mitzvah d’oraysa, the best bet is to try to wait and see if the storm passes and one can avoid the situation of safeik. The Magen Avraham writes to wait until midnight (but not later because, returning to our limud from Pesach, the latest time for the seder is midnight), but the Mishna Berurah quotes other Achronim who note that there is no formal shiur -- the amount of time to wait depends on one’s tolerance and that of one's familty, taking into account the mitzvah of simchas yom tov.
If the rain does not stop, kiddush should be recited in the sukkah and a k’zayis of challah eaten without a bracha of leishev ba’sukkah (to avoid a safeik bracha). If the rain lets up and one is still awake, one should go out to the sukkah, eat a k'beitza of challah (see the first comment to this post), and recite leishev ba'sukkah. One would therefore be yotzei the mitzvah of sukkah even according to the view that holds the k'zayis eaten in the rain does not count.
The only difference between the second night and the first according to Mishna Berura is that on the second night one can start the meal inside and eat a k’zayis afterwards in the sukkah. The reason we do the opposite on the first night (we start with the k’zayis in the sukkah) is to avoid having to add an extra bracha of she’hechiyanu, once at the start of the meal during kiddush and once at the end of the meal when eating the k'zayis and sitting in sukkah for the first time. On the second night this is not an issue because she’hechiyanu on sukkah was already recited the night before.
The Magen Avraham writes that on the second night one need not eat a k’zayis in the sukkah in the rain. This view seems easy to explain using the principle of sfeik sfeika: safeik whether it is Yom Tov or chol, and even if it is Yom Tov, safeik whether the halacha is like the Ran who requires eating k’zayis in the sukkah in the rain or not. (One might argue further that m’ikar hadin Yom Toc sheni is not even a real safeik but is vaday a weekday, but we treat it as Yom Tov because of minhag/takanah.) The Aruch haShulchan writes that the minhag follows this Magen Avraham, but again, the Mishna Berurah disagrees. Mishna Berura further writes that one should preferably wait before eating even on the second night.
(Please do not rely on this post for psak: see the S"A in siman 639 and the relevant Achronim. And hopefully this post will not be relevant l'ma'aseh anyway and we get some nice weather.)