Since it is forbidden to fast on Shabbat, do we fast on Yom Kippur if it falls on Shabbat?
Your premise is correct; all other fast days are postponed until Sunday when they fall on Shabbat. However, unlike all other fasts, Yom Kippur is not postponed, and is fully observed even on Shabbat. The Torah dubs Yom Kippur Shabbat Shabbaton—the “Shabbat of Shabbats,” implying that it takes precedence over Shabbat.
According to chassidic teachings, Yom Kippur falling on Shabbat doesn’t “deprive” us of the pleasures—eating, drinking, resting, etc.—which Shabbat normally affords us. Rather the extremely holy nature of Yom Kippur accomplishes the same objectives, albeit in a higher, more spiritual manner.
Here are two explanations on this topic, culled from the chassidic works:
1) King David says, “Behold, G‑d’s eye is directed towards those who fear Him, to those who hope for His kindness, to rescue their soul from death and to sustain them in famine.” The Hebrew words for “to sustain them in famine”
(להחיותם ברעב) can also be translated as “to sustain them with hunger.” In a spiritual sense, “famine” refers to the soul’s yearning for closeness to G‑d, a yearning which derives from the fact that the soul is “a part of G‑d above,” and always desires to reunite with its Source. On Yom Kippur, when the soul and its needs and wants are bared, this hunger alone, the quest for spirituality, is sufficient to satiate and satisfy a person. On the holiest day of the year, we are fueled not by carbohydrates or proteins, but by the revelation of our very essence and its intrinsic relationship with G‑d.
2) The human’s physical need for nutrition stems from the soul’s need to be energized by the divine sparks inherent within every physical creation. This is because the soul has many levels, and only its lowest levels are normally expressed in the body, and these soul-levels require the spiritual nutrition derived from various foods. The essence of the soul, however, is far higher than these sparks, and therefore has no need to be fortified through their consumption. Thus, on Yom Kippur, when this essence is revealed and expressed within every Jew, there is no need for eating or drinking.
May we all experience a spiritually uplifting Yom Kippur, a Yom Kippur which will cast its holy glow—and have a concrete effect—upon the entire blessed new year.
Rabbi Naftali Silberberg