Gemara

74b (Mishnah): Tearing in order to sew [is an Av Melachah].

Question: Did they tear for the Mishkan?!

Answer (Rabah and R. Zeira): If worms made a hole in a curtain, they would tear it and sew it up.

105a (Mishnah): If one tears in order to sew two stitches [he is liable].

105b: The case is, the garment had a protrusion. (It must be torn to make it straight.)

(Mishnah): If one tore out of anger or due to his Mes, or did anything destructive, he is exempt.

Beitzah 32b (Mishnah): One may not break earthenware or tear paper.

This is because he is Metaken Maneh (fixes a Kli).

Rishonim

Rambam (Hilchos Shabbos 10:10): If one tears in order to sew two stitches, he is liable. If one tore to ruin he is exempt, for he is Mekalkel (destructive). If one tore out of anger or due to his Mes over which he must mourn, he is liable, for this calms his mind and Yetzer. Therefore, this is like Tikun. One who opens the neck hole of a shirt is liable.

Rambam (11): If one separates papers or hides stuck together, without intent purely to be Mekalkel, he is liable for a Toladah of Kore'a (tearing).

Rambam (23:6): One may not cut a reed tube, break earthenware or tear paper, for this is like Metaken Maneh.

Poskim

Shulchan Aruch (OC 340:13): One may not break earthenware or tear paper, for this is like Metaken Maneh.

Mishnah Berurah (40): He breaks or tears in order to put something on them (the resulting parts).

Mishbetzos Zahav (5): It seems that one who tears in order to fix is liable, like in Sa'if 14. It need not be in order to sew. If so, in Sa'if 13, why do we forbid because it looks like fixing? The Torah forbids! The Rambam is Mechayev one who tears to calm his anger!

Mishnah Berurah (41): This is when he is not particular about the size. If he is particular, this is also Mechatech (cutting).

Kaf ha'Chayim (85): If he is not particular about the size, it is forbidden mid'Rabanan.

Mishnah Berurah (41): If he tears paper in order to clean himself [after eliminating] or some other use, he is liable for tearing, like in Sa'if 14.

Shmiras Shabbos k'Hilchasah (23:16, and footnotes 54,55): If one has no cut toilet paper available, and there is no substitute he can use, for Kevod ha'Briyos one may tear it k'Le'acher Yad (unskillfully). I.e. he holds it with one elbow, and tears with the other elbow. It is better to tear off from a roll than to cut one piece into two, for the Bi'ur Halachah says that only the latter is a Torah Isur [when done skillfully]. He should try not to cut at the perforation, due to the Isur of cutting.

Nishmas Adam (29:2): The Rambam connotes that tearing paper is only Shevus. That entrire Perek discusses Shevus. Why isn't one liable for Kore'a? We must say that Kore'a applies only to something attached through weaving, sewing or glue. The Rambam is Mechayev one who separates glued papers for tearing. Tearing one paper is different. Perhaps tearing did not apply to their paper, which was from herbage. This requires investigation. If one separates hide into the two layers of which it is comprised, the Rambam (11:6) obligates for flaying. One who separates hides stuck together is liable for Kore'a (340:14). However, the Yerushalmi obligates for tearing hide. Even though the Rambam said 'it looks like Tikun Maneh', perhaps he is Mechayev. He says similarly about cutting a reed, and 146b connotes that one is liable for this.

Bi'ur Halachah (DH Ein): Sa'if 14 is Mechayev due to tearing. Why don't we say so in this Sa'if? Also, the Rambam connotes that it is mid'Rabanan. He taught this with other mid'Rabanan matters. The Mishbetzos Zahav left this difficult. One Acharon answered that tearing applies only to materials stuck together, like the curtains in the Mishkan, but not to one substance, e.g. hide or paper. This is wrong. The Yerushalmi explicitly says that tearing applies to hide. Nishmas Adam retracted due to the Yerushalmi.

Note: Nishmas Adam did not explicitly retract regarding paper. However, the Yerushalmi refuted his idea that tearing applies only to materials stuck together.

Bi'ur Halachah: I suggest a Chidush. Tearing in order to fix applies only in the middle, and he needs to fix both matters, like the curtains in the Mishkan. If worms made a hole in a curtain, they would tear it and sew it. The tearing is for the sake of the entire curtain. If a garment was sewn unevenly and had a protrusion, one who tears it is liable, for this enables fixing the entire garment. The Rambam obligates one who tears out of anger because he helps himself calm his anger. It is like a Tikun in the entire garment. This is unlike tearing something off the side of a garment in order to fix it, for it was too long or the edge was ruined. The piece torn off is not fixed. This is not called tearing in order to fix, rather, Metaken Maneh, for he fixes the garment. If it is a total Tikun, he is liable for Makeh b'Patish, for this completes the Kli. If it is not a total Tikun, it is mid'Rabanan. If one tears a piece from a whole page for use, this does not fix the page. It is detrimental to the page! This is not Kore'a in order to fix. The Rambam explains the Gemara "he is Metaken Maneh" to mean that it looks like he is Metaken Maneh. It is not a full Kli. Therefore, he taught this with mid'Rabanan matters. If one tears paper into many parts, and needs all of them, he is liable for Kore'a in order to fix.

Bi'ur Halachah: Mo'ed Katan 22b supports this. It asked why a verse needed to say that he tore into two pieces. This implies that Stam tearing is into two pieces, but not tearing from the side and throwing it out, and the garment remains. The Yerushalmi asked how Kore'a differs from cutting, and answered that Kore'a is in the middle, and cutting is on the side. I.e. Kore'a is when he fixes both sides. Cutting applies even when he cuts something off one side to even it. It says that cutting applies to one who weaves and cuts strands that stick out. However, perhaps it merely discusses a typical case. Sometimes one cannot tear on the side, for there is not enough to hold and tear.

Bi'ur Halachah: I needed this discussion to explain why the Gemara obligates for Metaken Maneh, i.e. according to the Rambam, who holds that opening the neck hole of a shirt is liable for tearing. It is not difficult for Rashi, who says that also that is due to Metaken Maneh. He holds that Kore'a does not apply when the Tikun is seen immediately after tearing. Therefore, we had to say that it is Metaken Maneh. Perhaps this is mid'Oraisa, due to Makeh b'Patish.

Bi'ur Halachah: The Rambam connotes unlike this. He exempts tearing to ruin, for this is Mekalkel. This implies that one is liable for any Tikun. However, perhaps since the part he tears off is wasted, this is called tearing to ruin. The Rambam (1:17) exempts one who digs and needs only the earth, due to intent for Kilkul, even though he benefits from the earth. In Halachah 1:7, he is Mechayev one who extinguishes a wick in order to save the lamp. Even though the extinguishing is Mekalkel, it fixes the lamp. This connotes unlike I said. Rashi calls this Soser in order to build elsewhere. Even if the Rambam holds like Rashi's Rebbeyim, who call this Soser in order to build in the same place, there is Tikun regarding the lamp, and Kilkul regarding the wick, which he discards. The first Tana obligates for this. All of this requires investigation.

Bi'ur Halachah (DH ha'Neyar): A letter (in which the paper is glued shut, like an airgram - PF) is designed to be read. Tearing it is Melachah ha'Tzerichah l'Gufah. One may not even tell a Nochri to open it. However, if a letter is enclosed in a second paper (an envelope - PF), like the custom, tearing the outer paper is Mekalkel [since it does not fix the envelope]. Even R. Yehudah exempts (73b). Therefore, a Nochri may tear it when needed, for this is Shevus of Shevus. However, Rashi holds that R. Yehudah is Mechayev Soser in order to build elsewhere, i.e. Mekalkel that is Tikun elsewhere. If so, this is mid'Oraisa, so one may not even tell a Nochri to do it. For a great need one may open a letter through a Nochri, especially if it is in an envelope. It is better to hint to the Nochri, without overtly asking him, like the Agudah says.

Chazon Ish (OC 61:2): If one tears an envelope without concern for the envelope, this is like tearing hide around a barrel. It is not Kore'a. However, there, on weekdays one unties it, and on Shabbos he tears. One tears an envelope also during the week, so perhaps this is not Mekalkel. For a Torah Kore'a, perhaps we do not rely on tearing destructively with intent only for the contents. If the torn envelope is useless, surely it is Mekalkel. Therefore, one may tear hide around a barrel even if he does the same on a weekday. If the torn envelope is useful, perhaps this is tearing of Tikun. If he intends to keep the torn envelope, perhaps this is a Torah Isur.