Gemara

48a (Rav Yehudah): If one opens the neck-opening of a garment, he is Chayav Chatas.

105b (Mishnah): If one tore out of anger or due to his Mes, or did anything destructive, he is exempt;

Contradiction (Beraisa): If one tore out of anger, or if a mourner tore [his clothing] due to his Mes, he is liable for Chilul Shabbos. Even so, he fulfilled his obligation to tear.

Answer: The Beraisa is R. Yehudah, who is Mechayev for Melachah she'Einah Tzerichah l'Gufah. The Mishnah is R. Shimon, who exempts.

Question: Is R. Yehudah Mechayev for something destructive?!

Answer #1 (R. Avin): It is considered constructive, for it calms his anger.

Answer #2: He did so to instill fear in his household;

Rishonim

Rambam (Hilchos Shabbos 10:11): If one separates papers or hides stuck together, without intent purely to be Mekalkel, is liable for a Toladah of Kore'a (tearing).

Maharil (Hilchos Yom Tov 37): A Yisrael may open a sealed letter on Yom Tov.

Poskim

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

Mishnah Berurah (41): All forbid tearing open a letter, even if he is careful to tear only the paper around the letters, and not the letters. (Note: It seems that the "letter" is a paper glued onto itself, like an airgram. The Bi'ur Halachah distinguishes between a Stam letter, and one enclosed in another paper, i.e. envelope.) One should not even tell a Nochri to do so, if not for a great need. The Agudah says to say to the Nochri 'I cannot read this as long as it is not open.' If the Nochri understands by himself and opens it, there is no problem.

Pri Chodosh (YD 118:18): If one tears paper covering an oven in order to open it, this is not Mekalkel. It is Tikun. It is a Torah Melachah. The Rambam says that separating papers or hides stuck together, without intent purely to be Mekalkel, is liable. Any Mekalkel who does not intend to ruin, e.g. he tears paper stuck on the oven not for Kilkul, rather, to open it to remove the food, this is Tikun for him. The Torah forbids it. One may not even tell a Nochri to do it. Many have the custom to ask a Nochri to open letters that come on Shabbos. We do not learn from a mistaken custom. Opening it is a Torah Isur. He intends for Tikun, to enable reading it. One should protest against those who do so. They will be punished for this.

Rebuttal (Eliyahu Rabah 307:32): One may tear the paper through a Nochri, for it is not intended to be permanent.

Nishmas Adam (29:2): There is no Tikun in tearing a letter. However, if one tears where the folds are stuck together through the Chosem (stamp of a signet ring), this is separating papers, for which the Rambam obligates. The Pri Chodosh says that the Rambam is Mechayev one who tears one paper. There is no source to say so. Also, most Poskim exempt Melachah she'Einah Tzerichah l'Gufah, so even separating papers is exempt. He says that the tearing is not Mekalkel, for he needs to open it, so all agree that he is liable. This is wrong. Every tearing is needed for something, and even so, R. Shimon exempts [in some cases].

Bi'ur Halachah (DH ha'Neyar): The Pri Chodosh holds that the Tikun is his need to open it. Sefer ha'Zichronos agrees. Chacham Tzvi (39) disagreed, and permitted through a Nochri. He did not explain why. It seems that he holds that it is Melachah she'Einah Tzerichah l'Gufah, for he does not improve the paper at all. Tosfos (94a DH Rebbi) says that this is why R. Shimon exempts one who tears out of anger to instill fear. However, if so, the Rambam, who is Mechayev for Melachah she'Einah Tzerichah l'Gufah, should forbid mid'Oraisa, and should forbid through a Nochri. Nishmas Adam says that all exempt. He holds that tearing over a Mes or due to anger is different. There, the tearing itself is the Tikun. (It calms his anger or instills fear.) Here, the paper is totally ruined, and the Tikun is for another reason. His reason and the Chacham Tzvi's are not clear. A letter is designed to be read, Its Tikun is ability to read it, so tearing it is Melachah ha'Tzerichah l'Gufah.

Bi'ur Halachah: The Magen Avraham (307:20) is stringent like Sefer ha'Zichronos. It seems that for a great need one may open it through a Nochri, especially if it is in an envelope. L'Chatchilah, it is better to do like the Agudah says. (He just hints to the Nochri.)

Kaf ha'Chayim (307:104): Eliyahu Rabah (307:32) and Aruch ha'Shulchan bring Maharil, who permits a Yisrael to open it. If the Nochri does not understand to open it, one may overtly tell him to.

Chazon Ish (OC 61:2): Binyan and Stirah do not apply to a letter (airgram). The Melachah of Kore'a applies. This is not destructive, for he is particular not to ruin the letter. Perhaps it is mid'Oraisa, like cutting the neck opening of a shirt, according to the Rambam. The Mishnah Berurah explains that Rashi holds that Binyan is not the Melachah of Kore'a. The Shevus of Metaken Maneh applies. In any case it is a Safek Torah due to the Rambam's opinion, so one cannot be lenient.

Shulchan Aruch (14): If one separates papers or hides stuck together, not with intent for pure Kilkul, he is liable for a Toladah of Kore'a.

Magen Avraham (18): If papers became stuck together through wax or at the time of binding, one may open them, like above 314:10, when it was not made to be permanent. All the more so here it is permitted, for it happened automatically.

Mishnah Berurah (43): If he intended for any benefit, even without intent to stick them together again, he is liable. Kore'a does not require intent to sew. Intent for any Tikun suffices so that it is not considered Mekalkel. If he intends for Kilkul he is exempt, like all Mekalkelim.

Mishnah Berurah (44): All the more so, one who tears on a seam in a garment, for some benefit, is liable.

Mishnah Berurah (45): If papers were stuck together in the place of the letters, one may not separate them [due to erasing].

Shmiras Shabbos k'Hilchasah (9:10,12): If a container (e.g. of yogurt) is covered with paper glued on, on Shabbos one may tear it off destructively [without any intent to use the paper], or tear the paper itself. He may not tear letters or pictures.

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! Maharalbach (79) says that one who tore a Sidur is liable according to the Rambam, who is Mechayev one who tears to calm his anger! This requires investigation.

Bi'ur Halachah (DH v'Chayav): I say that one who tore a Sidur is liable for Makeh b'Patish. [After binding,] bookbinders look for pages stuck together, and separate them. Rashi holds that when the Tikun comes immediately, this is not Kore'a, rather, Makeh b'Patish. Also Tif'eres Yisrael (Kalkeles Shabbos 38, ha'Makeh b'Patish) says so. Perhaps the Rambam agrees here, but regarding opening a neck hole, he holds that this is not the final Melachah.

Note: Maharalbach discussed one who tore a Sidur in anger. He said that the Rambam holds that he is liable, for he calmed his anger. It seems that the Bi'ur Halachah never saw the Teshuvah. He assumed that he did not tear in anger, for if so, we need not learn from Maharalbach that the Rambam is Mechayev. The Rambam explicitly says so! The Bi'ur Halachah concluded that the man separated pages. (Really, Maharalbach discussed whether the man is exempt from paying because he was Chayav Misah. Perhaps Mishbetzos Zahav cited this to show that Maharalbach gave weight to the Rambam's opinion, even though most Poskim exempt Melachah she'Einah Tzerichah l'Gufah.) Tif'eres Yisrael merely said that opening a neck hole is liable due to Makeh b'Patish. Tif'eres Yakov explains that he does not obligate for Kore'a, for one is liable only in order to sew!

Kaf ha'Chayim (90): One may not even tell a Nochri to cut them. One who buys a new Sefer, or bound a Sefer, should separate the pages before Shabbos if he might need the Sefer on Shabbos.