The Gemarrah on 9a says "Azei Succah asuirin kol shivah". The Shulchan Aruch rules in accordance with the Rishonim that hold this prohibition applicable to the walls of the Succah as well. According to this ruling how are "car succahs" permissible? A car succah is made by opening the front and back doors as your 1st 2 walls using luvud to extend them to the grond. The 3d wall is the other side of the car using the law of "dofen akumah" assuming that the width of the car is less than 4 amos which all cars are. the s'chach is then placed on the 2 open doors making a presumably kosher succah assuming it has the requisite 7x7 tefach measurement. Back to my question: If the walls of the Succah are "Muktzeh" (maybe even Muktzeh D'oraisuh) how can you take apart you car succah and close the doors to your car? You are taking the walls of a Succah and using them for another purpose!! (unless you drive with the car doors open; please don't try that at home.)
The only possible answer I can think of is very difficult to understand and probably does not apply to our situation. The Be'ur Halachah says (sorry I don't have the citation in front of me; he says this when dealing with "Muktzeh" of the decorations in a succah) that the decorations don't have a Din of Muktzeh until "Bein Hashmushos". Until one Bain Hashmushos period passes the decorations are not Muktzeh. Firstly, I don't understand this Halachah (in Hilchos Shabbos, something can become "Muktzeh" during the day of Shabbos). Once we understand this Be'ur Halachah maybe we can apply it to our "car Succah" and make it permissible.
You are asking a very important and observant question.
The "Atzei Sukah" become Asur and are Muktzah for all seven days because the Etzim have been designated for use in the Sukah at the time that Yom Tov enters (i.e. Bein ha'Shemashos of the first evening of Sukos). One may not make a Tenai ("I am not exclusively designating this material for the Sukah, for I retain the ability to use it during Bein ha'Shemashos") before Yom Tov to use the Etzim later on Sukos for personal use (as one may make in order to use the Noyei Sukah), since it is not possible to use the Etzim the first day of Yom Tov during Bein ha'Shemashos, because removing them from the Sukah would involve a Melachah on Yom Tov (Mishnah Berurah OC 638:6; Beitzah 32a).
However, a Tenai may be used for a Sukah made during Chol ha'Mo'ed, such as a "car Sukah," even if it is used as a Sukah during Bein ha'Shemashos. Furthermore, it is logical to assume about something that one obviously does not intend to leave as part of the Sukah, that it is understood he is using it with a Tenai and the item does not become Asur. (The Mishnah Berurah writes a similar Sevara from the Taz, regarding fancy lace wall-coverings which one does not intend to leave hanging at night.) Only if Yom Tov (or Shabbos) comes when the material is being used as a Sukah will the material then become Asur (since it is not possible to use it during Bein ha'Shemashos because of Meleches Shabbos or Yom Tov).
The Bi'ur Halachah you referred to (OC 638, DH Asur l'Histapek) says that a Sukah decoration does not become Asur until Bein ha'Shemashos comes (and one has made no Tenai), even if it was hung up on Chol ha'Mo'ed. (This is because the decoration does not become a part of the Sukah until it is there at the beginning of a new day. It presumably should not apply to an item that actually comprises the Sukah itself, and is not just a decoration.)
As for an item that becomes Muktzah in middle of Shabbos, the Halachah is that an item does not become Muktzah for an entire Shabbos if it is becomes Muktzah in the middle of the day (Muktzah l'Chatzi Yom; SHULCHAN ARUCH 310:3).
Best wishes, Mordecai