Answer: The Shulchan Aruch (OC 319:1) lists three conditions for allowing one to select an item from a mixture on Shabbos so as not to transgress the melacha of borer. One must select the ochel, wanted item, from the pesoles, unwanted item, use one’s hand rather than an implement, and it must be for immediate use. While peeling requires one to remove the pesoles from the ochel, the Rema (OC 321:19) writes that one may peel garlic and onions on Shabbos providing it is for immediate use. The Biur Halacha (321) explains that this is permitted because it is the normal way of eating such food.
The Magen Avraham (OC 321:30) extends this halacha to peeling apples. R’ Moshe Feinstein (Igros Moshe OC 1:124) notes that as it isn’t feasible to peel an apple without a knife, doing so is considered as an extension of one’s hand, rather than considered to be using a special implement.
The Pri Megadim (Eshel Avraham 321:30) challenges the Magen Avraham asking why one can’t peel apples even for later, writing that as both are edible, peeling an apple would be like cutting it in half (See Rivevos Ephraim 8:118:8). Thus, the Aruch Hashulchan (OC 319:22) writes that peeling does not constitute borer and thus allow one to use a peeler on Shabbos. Likewise, the Kaf Hachaim (OC 321:141) allows one to peel apples for later as the peels are edible.
Nonetheless, R’ Moshe Feinstein (Igros Moshe OC 4:74 Borer 8) writes that the Mishna Berura (321:84) seems to side with the Magen Avraham, and while apple peels may be edible, if one is discarding them, then they are considered to be pesoles. As the peeler acts as a kli for borer, one cannot use it on Shabbos (See Machazeh Eliyahu 1:51).
R’ Yehoshua Neuwirth (Shemiras Shabbos Kehilchasa 3:30) differentiates between edible and inedible peels. When the peel is considered to be edible, then the peeler can be used as it is like a knife that cuts two pieces. It may not be used to remove inedible peels, however, as it is considered to be a kli for borer.
In conclusion, one may not use a peeler for inedible peels on Shabbos. While some poskim allow one to use one to peel edible peels, one should ideally only do so if one isn’t going to discard the peels.