If someone passes away on the first day of Yom Tov - should the burial be done on Yom Tov Sheni? How is it done?
The gemara (Beitza 6a) rules that burial on yom tov sheni is permissible, and Rambam and Shulchan Aruch rule like the opinion in the gemara that even things that are not strictly necessary for burial may be done if they are in the spirit of kavod hames, since yom tov sheni is treated as a weekday with regards to burial.
However, nowadays, when we have the ability to postpone burial due to refrigeration, some poskim (notably, R' Moshe Feinstein) are of the opinion that we should be stringent and wait until after yom tov, since holding a funeral on yom tov may cause its unnecessary violation by ignorant or irreligious people
From what I've heard (need sources), the rabbis were very lenient with the second day of yomtov when it came to prompt burial; the human dignity of not leaving a corpse out decomposing had a very high value.
However, today, when the corpse can be refrigerated, I've heard the common practice is to wait until after all of yomtov before the burial.