Once a sukka has been built, it is set aside for the holiday and should not be used for other purposes.
This is the conclusion of Rav Sheshet in the name of Rabbi Akiva, who points to the passage Hag ha-Sukkot shivat yamim la-Shem – the holiday of Sukkot is seven days for God (Vayikra 23:34) – and its interpretation as given by Rabbi Yehuda ben Beteira. He compares the word hag to the sukka, teaching that just like the hagiga sacrifice (Festival peace-offering) belongs to God, so the sukka belongs to God.
There are a variety of opinions about how to define the prohibition in this case. Tosafot suggest that the pasuk actually confers a level of holiness on the structure of the sukka, so it is forbidden to use, just as kodashim – things belonging to the Temple – cannot be used. Others compare it to the standard rules of muktze that we are familiar with from the laws of. The wood used for the sukka has been set aside for a specific mitzva purpose, so it cannot be used for other purposes.
Connected to this discussion is whether the prohibition applies to all parts of the sukka or only to the main parts of it – that is to say, the minimum needed for the sukka to be kosher (this is a disagreement between the R”i and Rabbeinu Tam quoted in Tosafot) – and whether it only applies when the sukka is standing or even if it falls down over the course of the holiday (see the discussion in the Me’iri and the Rosh).
Finally, some distinguish between different component parts of the sukka. The skhakh – roofing – may be seen as an issue of kodashim, the walls as a question of muktze, while use of the decorations might be perceived as making a mockery of a mitzva.
As far as the halakha is concerned, according to the Shulḥan Aruk (Orah Hayyim 638:1) it is forbidden to take even a chip from the sukka to use as a toothpick.