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A. Shemoneh Esrei:
In Shemoneh Esrei one must remember to add the following statements: Zachreinu; Mi Kamocha; Hamelech Hakadosh; Hamelech Hamishpat; Ukesov; Ubesefer, Oseh Hashalom. For further details on the laws relating to Shemoneh Esrei see Halacha 5.
Veata Chonantanu: On Motzei Rosh Hashanah one needs to recite Havdala [Ata Chonantanu] in the blessing of Chonen Hadaas, just like on Motzei Shabbos. [However when Motzei R”H falls on Friday night it is not recited-see next Halacha!]
If one forgot Ata Chonantanu in Maariv of Motzei Yom Tov what is the law?
It follows the same ruling as one who forgot to say it on Motzei Shabbos. Thus if he ate or drank [liquids other than water] or did Melacha before saying Hamavdil, he must repeat Shemoneh Esrei.
B. Havdala on Motzei Yom Tov:
On Motzei Rosh Hashanah one is required to recite Havdala over a cup of wine just like on Motzei Shabbos. [However when Motzei Yom Tov falls on Friday night it is not recited-see C!]
Haeish: During Havdala of Motzei Yom Tov we do not say a blessing over fire.
Besamim: During Havdala of Motzei Yom Tov we do not say a blessing over Besamim.
Nussach: The Nussach of Havdala on Motzei R”H that falls on a weekday also contains the words “Bein Yom Hashevi Lesheishes Yimei Hamaaseh”, even though it is now in middle of the week [and it thus seems irrelevant to mention this statement]. [Thus it follows the same Nussach as any Motzei Shabbos.]
Are the Pesukim of Hinei Keil Yeshuasi recited on Motzei Yom Tov?
May one who did not say Havdala on Motzei Yom Tov say it the next day?
One who did not say Havdala on Motzei Yom Tov is to say Havdala the next day [until sunset], and is not to eat or drink anything, besides for water, until he does so. If one did not say Havdala the next day [prior to sunset] then he may no longer say Havdala, and may thus continue eating and drinking as usual.
If one forgot to say Havdala on Motzei Rosh Hashanah and then remembered on Tzom Gedalia what is he to do?
He should say Havdala on Tzom Gedalia and give the wine to drink to a child which has reached the age of Chinuch.
Is Vayiten Lecha recited on Motzei Yom Tov?
 Piskeiy Teshuvos 294/4
 491/1; 602/2; Michaber 601/1 [Regarding Motzei R”H]
 The reason: As the blessing over a flame was only instituted to be said on Motzei Shabbos and Motzei Yom Kippur in which case fire was prohibited for the entire day and it now returns to be permitted, as explained in 298/1-2. However on Yom Tov fire was never prohibited when being used for a need on the Holiday. [Admur ibid]
 The reason: One is only required to say a blessing over Besamim on Motzei Shabbos as it is done in order to calm the soul which is pained by the leave of the extra soul it received on Shabbos and has now exited from it on Motzei Shabbos. However on Yom Tov there is no extra soul given at all. [Admur ibid]
Background and analysis:
“At all” so adds Admur in 491/1. This follows the ruling of Tosafus Pesachim 102a; Rashba 3/290; Radbaz 2/620 that there is no extra soul on Yom Tov. However the Rashbam Pesachim 102b rules there is an extra soul on Yom Tov, and so is the ruling of the Zohar. Vetzaruch Iyun how the above Poskim can go against the Zohar. Perhaps then in truth there are many levels of an extra soul, and on Shabbos one receives a higher level than on Yom Tov. However Tzaruch Iyun from Admur which states “on Yom Tov there is no extra soul at all”. In truth in the source of Admur from the Magen Avraham and Levush these words “at all” are omitted. Furthermore in certain prints of the Shulchan Aruch of Admur this entire Halacha is missing. It most probably is the case then that Admur did not write this word and it was written by the copier. [Shaar Hakolel 17/22; Likkutei Sichos 31 Ki Sisa] However the Rebbe in Likkutei Sichos 31 Ki Sisa footnote 15 explains that based on Nigleh there is no extra soul at all on Yom Tov, and it is only based on Kabala that there is an extra soul, hence Admur wrote in his Shulchan Aruch, which follows the rulings of Nigleh, that there is no extra soul “at all”.
Another reason: Regarding the second reason mentioned by the Bach for smelling Besamim, due to the return of the fire of Gihenim, the Mordechai [Pesachim 105] rules that on Yom Tov Gihenim is not subdued, and thus there is no need for Besamim on Motzei Yom Tov. However Tosafus [Beitza 33b] rules that Gihenim is subdued on Yom Tov and hence the reason for the omission of Besamim is because there is no extra soul.
 As explained in 296 [ibid; Vetzaruch Iyun as in 296 no mention of this is made. See also 473/9]
 The reason: As one is simply stating the order of Havdalos that are written in the Torah “Bein Kodesh Lechol, Bein Or Lechoshech, Bein Yisrael Leamim etc”. [ibid; See also 599 regarding Vetodieinu for a similar ruling]
 Peri Megadim 491 M”Z 1; and so is the custom of the Rebbe as was witnessed in public.
 Mateh Efraim 601/10; 624/5
 For list of opinions on this question see Sdei Chemed Asifas Dinim Mareches Hei 15; Kaf Hachaim 299/24
 So rules Kol Bo 59; Rav Akiva Eiger 299/6; Beis Efraim; Mishneh Berura 299/15; Piskeiy Teshuvos 299/4; SSH”K and so infers Sdei Chemed [Asifas Dinim Mareches Heim 15] from Admur 299/8 which allows saying Havdala of Shabbos the next day even if one purposely skipped Havdala at night, thus proving that the next say is not a Din Tashlumin, but rather a continuation of the obligation.
Other Poskim: Many Poskim rule that one may not make up Havdala even the next day, as Safek Brachos Lehakel. [Sdei Chemed himself ibid; Chida in Birkeiy Yosef 491/1; Machazik Bracha 491/1; Chesed Leavraham 491/2; Ben Ish Chaiy Vayeitzei 23 and many other Poskim listed in Sdei Chemed ibid]
 However past sunset he is to no longer say Havdala as it is now a question whether or not the next day has begun, and many Poskim even hold that Havdala on Motzei Yom Tov may not be made up even the next day. Thus after sunset one is to be stringent. [So also rules Piskeiy Teshuvos 299/10]
 299/8 regarding Havdala after Shabbos, and the same rule applies to Motzei Yom Tov
 So is understood from all the Poskim mentioned in the first footnote which only extend the allowance for the next day.
Other Opinions: The Beis Yehuda 2/28, and other Poskim [brought in Sdei Chemed ibid] rule that one may say Havdala the entire week. We do not rule like this opinion, as brought in Chachma Umusur [see Sdei Chemed] that all the Poskim argued on his ruling. Likewise the Chida [Birkeiy Yosef 491/1; Machazikei Bracha 491/1] argues against his ruling.
 Sdei Chemed Asifas Dinim Hei 15