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The Gemara in Erchin 15b compares Lashon Hara to the Big Three:

תנא דבי רבי ישמעאל כל המספר לשון הרע מגדיל עונות כנגד שלש עבירות עבודת כוכבים וגילוי עריות ושפיכות דמים כתיב הכא לשון מדברת גדולות וכתיב בעבודת כוכבים (שמות לב, לא) אנא חטא העם הזה חטאה גדולה בגילוי עריות כתיב (בראשית לט, ט) ואיך אעשה הרעה הגדולה הזאת בשפיכות דמים כתיב (בראשית ד, יג) גדול עוני מנשוא גדולות אימא תרתי הי מינייהו מפקא

The Academy of R' Yishmael taught: Anyone who speaks Lashon Hara, his sins are expanded to be comparable to three sins: idolatry, immorality, and murder. It is written here [by Lashon Hara] (Tehillim 12:4), "A tongue speaks great things," and it is written by idolatry (Shemos 32:31), "Please, this nation has sinned a great sin." By immorality it is written (Bereishis 39:9), "And how can I do this great evil thing?" By murder it is written (Bereishis 4:13), "Is my sin too great to bear?" [Asks the Gemara:] I might say, "great things" [indicates] two [and not three]! [The Gemara answers:] Which of them will you remove?

The Gemara in Sanhedrin 74a rules:

כל עבירות שבתורה אם אומרין לאדם עבור ואל תהרג יעבור ואל יהרג חוץ מעבודת כוכבים וגילוי עריות ושפיכות דמים

All sins in the Torah, if they say to a person, "Violate and you will not be killed," he should violate and not be killed, except for idolatry, immorality, and murder.

If Lashon Hara is compared to these three sins, why isn't one obligated to give up his life for Lashon Hara (under normal circumstances)?

As I noted in my answer to a very similar question, R. Isaac Ben Sheshet has a responsum in which he asserts that the Sages often exaggerated the severity of sins in order to prevent people from stumbling in them. He specifically gives this Talmudic passage about lashon harah as an example of such an exaggeration:

Shu"t HaRivash # 171

ומה שאמרת שהגאונים והרשב"א ז"ל כתבו שהעובר על החרמים והשבועות הוא בכלל לא ינקה והיינו כעובד ע"ג דכתיב בה ולא ידבק בידך מאומה מן החרם ע"כ אם הפליגו להגדיל העון ההוא כמו שעשו חז"ל שאמרו שבכל עברות שבתורה נאמר בהן ונקה וכאן לא ינקה וכן שנפרעין ממנו ומכל העולם לא אמרו בו שיהרג ואל יעבור כמו באותן ג' עברות וזה לא עלה על דעת שום אדם ולא חשב אנוש אלא שדרך החכמים להפליג בהגדלת העונות כדי שישמר אדם מהכשל בהן אמרו בפ' יש בערכין כל המספר לשון הרע מגדיל עונות כנגד שלש עברות ע"ג וגלוי עריות ושפיכות דמים וכו' וכן במס' נדרים יצא ר' עקיבא ודרש כל שאינו מבקר את החולה כאלו שופך דמים ובפ' ר"א דאורג אמרו הקורע בגדיו בחמתו וכו' יהיה בעיניך כעובד ע"ג היאמר אדם באלו וכיוצא בהם יהרג ואל יעבור הא ליכא למימר

Thus, one is not obligated to give up his life rather than transgress lashon harah because the comparison to the three sins for which one does have to give up his life was just an exaggeration.

Rav Chaim Kanievsky (Doleh U'Maskeh pg.263) was asked this question. He answered that lashon harah is a very grave sin, but one is not chayav misah. In footnote 29 the Orchos Tzadikim shaar Lashon Harah is brought as a source. The Orchos Tzadikim explains why lashon harah is equated with the big 3 sins. See also Shaarei Teshuvah 3:202.

Orchos Tzadikim (Sefaria translation):

Our Rabbis, of blessed memory, said, "If one speaks gossip, it is as though he denied God" (Arakin 15b, and see T.P. Peah, 1:1). As it is said (Ps. 12:15), "Who have said : 'Our tongue will we make mighty : our lips are with us : who is lord over us?' " (Arakin 16b, and see T.P. Peah, 1:1). And therefore our Sages considered him as though he denied God, for he does a great evil to his companion, by making him odious in the eyes of the people and causing him other losses, though he himself derives no profit out of this. There is no doubt that one who makes a habit of speaking gossip has thrown off the yoke of Heaven from upon him, for he sins without any benefit to himself, and he is worse than a thief or an adulterer, for they pursue after their pleasure (Shohar Tov, 120:3). And there is no throwing off the yoke of Heaven as bad as when one deals in gossip. Our Sages further said, "The sin of gossip is weighed equally with the sin of idolatry and sexual immorality and bloodshed" (Arakin 15b). Now it is very astonishing that gossip should be equated with these sins, which are those to avoid which a man must rather let himself be killed than transgress (Sanh. 74a). And they said, "The sin of idolatry is so great that anyone who confesses to it, it is as though he denied the whole Torah" (Hullin 5a). And they said, "One who lusts after any form of idolatry, is a rebel against the whole Torah" (Hullin 4b).

And it is important to give the full meaning in this matter, for a gossip repeats his folly; ten times or more every day he humiliates and shames people, aside from the damage that he does to the one that he speaks against. And even a small transgression, when done many times, becomes great, just as although a single hair is soft and very weak, if you gather many hairs together, you can make of them a strong rope. And when they said that gossip is on a scale with those three mortal sins, they referred to one who commits any of these sins once, because of great temptation, but not to one who is such a complete apostate that he has excluded himself from the people of Israel, in order to commit these sins repeatedly. Moreover, a gossip finds it difficult to repent, because he is used to this habit and has taught his tongue to speak evil. Furthermore, this sin appears very light in his eyes, for he says, "I did not do anything — it was just talk." He does not consider the great damage he does, and therefore he does not repent. And even if he should repent, his repentance is not complete, for he does not realize the enormity of the sin which he has committed. Moreover, he must first obtain forgiveness from those against whom he has spoken, and he cannot remember whom they all are. And it may happen that he spoke against a man, and did him evil, and caused him harm and forgot what it was that he said about that man, for gossip is always covered up; it is a blow struck in secret. The gossiper is here and smites with his tongue a person who is far away from him (Arakin 15b, Gen. Rabbah 98:19). And this type of sinner is ashamed to let his victim know that he has done him evil. Sometimes he speaks about a defect in the family of the object of his gossip, thus injuring the generations that come after him, and there is no forgiveness for this, for our Sages said, "For one who speaks about a flaw in a family, there is no forgiveness eternally" (T.P. Baba Kamma 8:10).