Gemara

73b - R. Yochanan: Chetzi Shi'ur is forbidden mid'Oraisa;

Reish Lakish: It is forbidden only mid'Rabanan.

Question (Mishnah): If one swore not to eat Neveilah or other Isurim and he transgressed, he is liable (even) for the oath.

Why does the oath take effect? Neveilah is already forbidden by the oath of Kabalas ha'Torah at Sinai!

Answer #1 (Rav, Shmuel and R. Yochanan): The case is, he included Heter (permitted food) in the oath. Since the oath takes effect on the Heter, it also takes effect on Isurim.

Answer #2 (Reish Lakish): According to Chachamim, who say that a Stam (unspecified) oath forbids a full Shi'ur, we can say that he specifically forbade Chetzi Shi'ur;

According to R. Akiva, who says that a Stam oath forbids even Chetzi Shi'ur, the oath takes effect even if he swore Stam.

Rishonim

Rambam (Hilchos Shevisas Asor 2:3): If one ate or drank less than a Shi'ur (on Yom Kipur) he is not Chayav Kares, even though it is forbidden mid'Oraisa. He is lashed mid'Rabanan.

Source (Magid Mishnah): The Halachah follows R. Yochanan against Reish Lakish. We learn lashes from a Kal va'Chomer from one who transgresses a Lav mid'Rabanan.

Lechem Mishneh: Even though the Rambam rules like R. Yochanan, he also rules that a Shevu'ah takes effect on Chetzi Shi'ur, like Reish Lakish. This is like the conclusion (Shevu'os 23b) that oaths take effect on an Isur which is not explicit in the Torah.

Question (R. Mano'ach): Since R. Yochanan learns Chetzi Shi'ur from "Kol Chelev...Lo Sochelu", one should be lashed mid'Oraisa for it! If he is not lashed, what practical difference is there between R. Yochanan and Reish Lakish?

Note: The Mishnah Berurah (611:3) says that if Chetzi Shi'ur is mid'Oraisa we must be stringent about a Safek. Presumably, R. Mano'ach holds like the Pri Chodosh (613:7) who says that since there is an Asmachta for the Isur, we are stringent about a Safek.

Poskim

Shulchan Aruch (OC 612:1): If one eats the amount of a large date on Yom Kipur he is liable.

Taz (1): The Tur (and Shulchan Aruch) only bring Halachos that are relevant today. The Shi'ur for Chiyuv is relevant for a sick person who must eat on Yom Kipur; if possible he should not eat enough to join for a Shi'ur.