Hashem says קשה עלי פרידתכם (Bamidbar Rabba 21:22 about Shemini Atzeres). The Imrei Noam translated it, "It is painful for me when there is machlokes and division amongst you. It is especially painful for Me when the machlokes is עלי, about Me – when you claim you are fighting for My honor."
Throughout Jewish history, there were many disputes — among individuals and communities – and the motives were often the preservation and continuity of Yiddishkeit and to stand up for Hashem's honor. At times those fights were indeed important, and thanks to those disputes, our adherence to Torah and mitzvos continues to the present day. However, sometimes, people thought they were fighting leshem shamayim, but it was just the yetzer hara.
Likely, Korach thought his ambitions were altruistic, fighting for Klal Yisrael's honor or his spiritual growth. However, had he looked closer, he would have discovered that behind his machlokes was jealousy and a drive for glory. He would have discovered that what he thought was righteous indignation was the yetzer hara.
The Baal HaTanya writes, "Most tzaros come from machlokes 'leshem shamayim.' May Hashem save us from them."
Kayin and Hevel had a dispute, which ended with Kayin killing Hevel. What was the disagreement about? The Midrash (Bereishis Rabba 22:7) states two opinions. One is they debated on whose property the Beis HaMikdash should be built on, each wanting the Beis HaMidash on their property. A second explanation is that they were arguing over the ownership of the earth, who owns the land, its elements, and substances.
"Kayin and Hevel said, 'Let's divide the world between us' [and they did so]: One got the land, and the other became the owner of all mitaltelen (elements/substances). The landowner said, 'You are standing on my land! Get off' "The other replied, 'And you are wearing my clothing. Remove the clothes.' 'And you must fly [in the air].'" This was the spirit of the dispute, which ended with murder.
Baalei mussar say that both explanations of the Midrash are, in fact, telling the same story. The real impetus of the machlokes was wealth. But they were ashamed to fight over such petty issues, so they turned it into a fight over who would host the Beis HaMikdash. But the core of the machlokes was money.
Reb Yohonasan Eibshitz, zy'a, (Yaaras Dvash, drush 8) writes, "In every machlokes, the yetzer hara assures us that our intentions are purely leshem shamayim, and chas veshalom even to suggest otherwise! So, how can we recognize whether a machlokes is leshem shamayim? This is the test: If both parties love each other with all their heart and soul aside from the issue they are arguing about, that is a sign that their argument is leshem shamayim. However, if they hate each other, the machlokes isn't leshem shamayim. The Mishnah hints at this when it states, איזו היא מחלקת שהיא לשם שמים, which can mean, how do we know that a machlokes is leshem shamayim? The answer is זו מחלקת הלל ושמאי, if it is similar to the machlokes of Shamai and Hillel, for Shamai and Hillel loved and honored each other. This proves that their machlokes was leshem shamayim. But Korach and his followers harbored animosity towards Moshe. They nearly stoned him. This proves that their dispute wasn't leshem shamayim. With this test, one can know whether his intentions are leshem shamayim or not."
The Bnei Yissaschar zt'l gives us another sign to help determine whether your motivation is leshem shamayim or jealousy. Generally, when fighting with "religious indignation," one’s emotions are passionate. He feels like an exploding volcano, ready to knock down all people who are going against Hashem. Therefore, he should gauge himself and see whether he possesses such furor when performing a mitzvah, such as tallis and tefillin, succah, helping the poor, etc. If this passion surfaces only by machlokes, that's an indication that it is inspired by the yetzer hara, not the yetzer tov.
The Imrei Noam zt'l taught that the worst sins can be committed when one thinks he's motivated by spirituality. No G-d-fearing person would think of destroying someone's life or parnassah. But when the yetzer hara convinces a person that he must take a stand to avenge Hashem's honor or to preserve Yiddishkeit, and so on, and that he must fight with an individual or a community, he becomes liable to do sins he would never consider doing otherwise. If he is wrong, and Hashem doesn't want the machlokes, he is taking Hashem's name and using it as a powerful weapon to do terrible things.
The Imrei Noam writes that this is hinted at by the words, יצרו של אדם מתגבר עליו כל יום ואלמלא הקב"ה עוזרו לא יכל לו (Succah 52:). This can be translated, אלמלא הקב"ה עוזרו, if it weren't for the yetzer hara's claim that it must be done for Hakadosh Baruch Hu's honor, לא יכל לו, the yetzer hara couldn't have convinced the person to do such awful deeds.
Another problem with a machlokes “leshem Shamayim” is that such a machlokos can last a very long time. This is because the instigators don't repent, for they think they're acting righteously. While a dispute over mundane matters can be resolved over time, it can last forever when Hashem's name is involved. This is hinted at in the following Mishnah (Avos 5:17): כל מחלוקת שהיא לשם שמים סופה להתקיים, "A dispute that’s leshem shamayim will last." ושאינה לשם שמים אין סופה להתקים, "but a dispute that isn't leshem shamayim won't last."
There is a machlokes on how to pronounce אדנ"י, whether the emphasis is on the 'ד or the 'נ. Once, when this was being hotly debated, the Noda b'Yehuda commented, "Hashem says, 'My name that was written in holiness should be erased for the sake of peace.' Peace is more important to Hashem than His name. Surely Hashem doesn't want us to fight over the pronunciation of His name."