Question: Can I take the bins out on Yom Tov to be emptied? Can I bring them back in when they have been emptied?
Answer: The Shulchan Aruch (OC 308:35) writes that one is allowed to remove a chamber pot (graf shel rei’i) on Shabbos. Although rubbish is usually considered to be muktze, one may remove this due to its unpleasantness. Thus, R’ Yehoshua Neuwirth (Shemiras Shabbos Kehilchasa 22:45) writes that one may empty an overflowing or smelly bin on Shabbos. There is no need to carry it out with a shinui (ibid n111). R’ Moshe Feinstein (Igros Moshe OC 5:21:2) adds that whether rubbish is considered to be graf shel rei’i or not depends how bothered a person is by the unsightly rubbish. Likewise, R’ Ephraim Greenblatt (Rivevos Ephraim 1:232:2; 6:180) writes that as nowadays most people are bothered by overflowing bins, they can empty them on Shabbos.
R’ Moshe Feinstein (Igros Moshe 5:21:9) adds that while the Aruch Hashulchan (OC 308:60) writes that it is preferable for a non-Jewish person to remove it on Shabbos, one shouldn’t ask a child to do it, as they may grow up thinking that muktze items may ordinarily be moved.
R’ Ephraim Greenblatt (Rivevos Ephraim 7:303:1) writes that one is allowed to take the bins out to the street for pickup on Yom Tov, though it is ideal to do so before Yom Tov. The Steipler (quoted in Nitei Gavriel, Yom Tov 22:20) also allows one to remove rubbish bags and bins to the street on Yom Tov, as having a clean house is a necessity for Yom Tov.
While it would be ideal to leave the bin on the street until after Yom Tov, R’ Ephraim Greenblatt (Rivevos Ephraim 1:222:19) also allows one to bring the bin back into one’s yard on Shabbos or Yom Tov either if one needs it to put more rubbish in or because one is concerned that it will get stolen. The Chazon Ish held that as modern bins are made out of plastic or metal, etc. which aren’t absorbent, they don’t smell like ancient earthenware chamber pots and so don’t have the same muktze status when emptied (See Shemiras Shabbos Kehilchasa 22:45).
In conclusion, while it is preferable to place the bins out onto the street before Yom Tov and collect them in afterwards, one may take them out and bring them in on Yom Tov if necessary.