There is a Misva to decorate the Sukka before Sukkot; the Sukka decorations are referred to in Halachic literature as "Noi Sukka."
The Rama (Rabbi Moshe Isserles of Cracow, 1525-1572) writes in his glosses to the Shulhan Aruch (Orah Haim 638:3) that one may not move or handle the Sukka decorations on Shabbat or Yom Tob, as they are considered Mukse (listen to audio recording for precise citation). This ruling is cited in the name of the Tur (Rabbi Yaakov Ben Asher, Germany-Spain, 1270-1340). One must therefore ensure not to fiddle with the Sukka decorations, neither those which hang from the Sechach nor those attached to the walls. The Be’ur Halacha (commentary by Rabbi Yisrael Meir Kagan of Radin, 1839-1933) adds that if a decoration falls down on Shabbat or Yom Tob, one may not pick it up until after Shabbat or Yom Tob ends. One may pick it up on Hol Ha’mo’ed.
If a Sukka decoration falls on the table, and one wishes to remove it from the table, he may lift and shake the table so that the decoration falls. Alternatively, the Ben Ish Hai (Rabbi Yosef Haim of Baghdad, 1833-1909) rules that one may ask a three- or four-year-old child to remove the decoration from the table. It is forbidden, however, for an adult to directly remove the decoration from the table.
This Halacha does not apply to pieces of Sechach that fall onto the table. If chips of wood or bamboo, for example, fall onto the table on Shabbat or Yom Tob, one may remove them, because they have the Halachic status of "Geraf Shel Re’i" – a substance that people find revolting. Halacha permits removing such items even though they would ordinarily qualify as Mukse. Therefore, although one may not remove Sukka decorations from the table, it is permissible to remove pieces of Sechach that fell onto the table.
Summary: Sukka decorations are considered Mukse on Shabbat and Yom Tob, and one may not handle them on Shabbat or Yom Tob, even if they fall. If a decoration fell onto the table on Shabbat or Yom Tob, one may ask a young child (aged three or four) to remove it, or shake the table so it falls. If pieces of Sechach fall onto the table on Shabbat or Yom Tob, one may directly remove them from the table.
