The Rama (O.H. 295:1) writes that in Arvit of Motzaei Shabbat, the verse (Tehilim 90:17) “Vihi Noam” followed by Kedusha DeSidra (ie. ‘“Ve’ata Kadosh…”) are added after the Amida. The Tur (ibid.) explains that this is added to Arvit so as to delay the return of the sinners to Gehinam. The Ashkenazic custom is to omit this entire section if a Yom Tov falls on a weekday that following week.
The Sephardic custom common among Edot HaMizrah is that the entire section is recited, including “Shuva Hashem” etc. every Motzaei Shabbat, regardless of whether or not there is a Yom Tov that coming week. The latter opinion is based on the Zohar (Hakdamat Bereshit), which says that the “Yoshev Beseter Elyon” is an auspicious Mizmor which has the power to deflect negative spiritual forces, which are especially rampant after Shabbat is over. As such, it is recited every week due to its benefits.
Rabbi Shalom Messas (Shemesh Umagen, vol. III, § 85:6) and Rabbi David Ovadia (Nahagu Ha’am, Arvit, § 8) write that he Moroccan custom is that if a Yom Tov will fall during the coming week, the paragraphs of “Shuva Hashem ad Matai” and “Yoshev Beseter Elyon” are omitted, however the last verse “Orech Yamim Asbiehu” is said twice followed by Kedusha DeSidra. Interestingly, Rabbi Baruch Toledano (Kitzur Shulhan Aruch, § 273) agrees that this is the common custom, but writes that there were Mekubalim and great Torah sages who would recite the entire section starting with “Shuva”. The HIDA (Birke Yosef, ibid.) concurs but also testifies that he witnessed those that would recite “Shuva” quietly.
The Rama goes on to write that it is customary to make mention of Eliyahu Hanavi, especially in Havdalah, and to pray for him to herald the coming of Mashiah. The Gemara in Eruvin (43b) mentions that he will not come on Erev Shabbat or Yom Tov since people are preoccupied with preparations, or on Shabbat itself due to the violations of traveling beyond the boundaries of Shabbat. Therefore, as soon as Shabbat is over, we immediately pray for his arrival.
Summary: If Yom Tov falls on a weekday, Kedusha DeSidra begins with “Orech Yamim”, while “Shuva Hashem Ad Matai” and “Yoshev Beseter Elyon” are omitted, on the preceding Motzaei Shabbat.