When Yom Tov falls on Shabbos and Erev Sukkos is Erev Shabbos, when is one to bind the Lulav together, on Erev Yom Tov, Erev Shabbos, or keep them in the fridge and bind them Motzei Shabbos or Sunday morning?
According to Chabad custom, it is to be done specifically on Erev Yom Tov. This applies even in Eretz Yisrael. However, some in Eretz Yisrael do so on Motzei Yom Tov, or the next morning before Shacharis of the first day of Chol Hamoed.
It is forbidden to make a knot on Shabbos or Yom Tov, and therefore the Poskim write that those who make knots to bind their Lulav, as is the Chabad custom, must do so before Yom Tov. Now, in the Diaspora, this applies even in years that Yom Tov falls on Shabbos being that they keep a second day of Yom Tov on Sunday and therefore cannot do so on Sunday before the first shaking. This indeed was done by the Rebbe in the year 1980 when the first day of Sukkos fell on Shabbos that he made sure to bind the four species in the sukkah on erev Shabbos.
However, in Eretz Yisrael, one can technically suffice with making the knots on Motzei Shabbos, or Sunday morning for the first day of shaking, being that it is Chol Hamoed in which knot making is permitted, and so indeed rule some Poskim that if not for the issue of making the knots it would be better to bind the four species on the actual day of the mitzvah, which is in Israel on the first day of Chol Hamoed. Nonetheless, even they conclude that the custom still remains to bind the four species on Erev Yom Tov. This can be explained based on spiritual reasons, as records the Rebbe in a Mamar said in 1981 that binding the Lulav in the Sukkah on Erev Sukkos helps to emphasize the living status of the Sukkah, which we want to do prior to the start of the mitzvah of sukkah.
Sources: See Mateh Efraim 625:18; Ketzei Hamateh 625:20; Sefer Haminhagim Chabad p. 65; Mamar Parshas Naso 1981; Likkutei Sichos 22:159 footnote 55; Reshimos 4 p. 12; Otzer Minhagei Chabad p. 283; Piskeiy Teshuvos 651:1 footnote 4-5; Orchos Rabbeinu 2:249 that he did so on Motzei Shabbos