If a person passes away, Heaven forbid, on Ereb Shabuot, the burial takes place that day and the family members observe a brief period of Shiba, usually a few minutes to an hour or so. Once this brief period of mourning has been observed, the onset of Yom Tob terminates the remainder of Shiba. Ordinarily, mourners observe the Shiba mourning period for seven days, during which time they wear torn garments, do not bathe, and observe other laws. The onset of Yom Tob, however, terminates the Shiba period, and thus once the family members observed a few minutes of Shiba, they do not observe the remainder of Shiba after Yom Tob.
The onset of Shabuot affects the thirty-day observance of Sheloshim, as well. Normally, even after Shiba, mourners abstain from haircutting and shaving during the thirty days after the deceased’s passing. However, the Shulhan Aruch (Orah Haim 548:13; listen to audio recording for precise citation) rules that in the case under discussion, the brief period of mourning observed on Ereb Shabuot counts toward seven days of Sheloshim, and the first day of Shabuot itself counts as another seven. In the times of the Mikdash, people who could not bring their festival sacrifices on Shabuot were given another six days to do so, and thus the day of Shabuot can be considered a seven-day holiday. As such, this day itself counts as seven days of Sheloshim. Hence, if mourners observe a brief period of Shiba on Ereb Shabuot, they must observe only fifteen days of Sheloshim after the second day of Yom Tob. Thus, the onset of Shabuot not only discontinues Shiba, but also eliminates fourteen days of the Sheloshim observance.
It should be noted that the onset of Shabuot does not affect the traditional Arayat which is held after thirty days have passed since a person’s passing. The Arayat has to do with the soul of the deceased, and thus depends on the passage of thirty actual days. When it comes to the restrictions of Abelut, however, the period of mourning on Ereb Shabuot and Shabuot itself count for a combined fourteen days of Sheloshim, such that the mourners are required to observe only fifteen days of Sheloshim after the second day of Shabuot (or, in Eretz Yisrael, sixteen days after the one-day holiday of Shabuot).
Summary: If a person passed away and was buried on Ereb Shabuot, the family members observe a brief period of Shiba that day. After the holiday, they do not observe Shiba, and are required to observe only fifteen days of Sheloshim after the second day of Shabuot.