Hefker on Shabbos — Eruvin 71a
תלמוד בבלי מסכת עירובין דף עא/א
ואסתלוקי רשותא בשבת שפיר דמי
The Pri Chadash (Orach Chaim 444:2) writes that it is forbidden to make an object hefker (i.e. to renounce ownership of the object) on Shabbos or Yom Tov. This is also the opinion of the Ritva (to Shabbos 120a), who writes that it is forbidden because it resembles a kinyan (a transaction in which ownership is conveyed from the previous owner to a new one) which is forbidden on Shabbos as well.
On the other hand, Teshuvos Avodas HaGershuni () writes that it is permitted to make an object hefker on Shabbos (see also Peirush HaRosh al HaTorah to Shemos ). The Sha'ar HaMelech (Hil. Lulav 8:2) suggests that there is no real dispute. Rather, the Ritva refers to a case in which the person had in mind to make the object hefker specifically for another person to acquire it, as this certainly resembles a transaction. However, where a person's intention is only to divest himself of the object, hefker does not resemble a transaction, and is permitted (see also Aruch LaNer to Sukkah 34b d"h
Teshuvos Reishis Bikkurim (§3) cites our Gemara as evidence that it is permitted to make something hefker on Shabbos, as we see here that although a person cannot acquire dominion over a domain on Shabbos (as this resembles a transaction — Rashi d.h. U'Mikna), it is permitted to renounce dominion over a domain on Shabbos.
R' Yosef Engel (Gilyonei HaShas here) rejects the proof. He explains (on the basis of an expanded analysis that he wrote elsewhere) that hefker is not a renunciation, but a conveyance of ownership from the original owner to the entirety of the Jewish people! Hence, it is like a gift — which resembles a transaction, and is forbidden on Shabbos — and not comparable to the "pure" renunciation of dominion that is the subject of our Gemara.