From another website, lekavod his yahrtzeit:
The Mishna in Pirkei Avos says that a “Machlokes L’Shem Shamayim Sofo Lihiskayem.” This seems strange. If the argument is really for the sake of Hashem why will it last forever? The mishna should say that it will eventually be resolved.
It is said in the name of Rav Yisroel Salanter that sometimes when arguing, people tend to hide behind the cloak of religion. Rather than admitting that their position is based on their own personal interest they say they are taking a stand “L’Shem Shamayim”. Therefore the Mishna says if one admits that his position is really self interest he can eventually compromise and the dispute can be resolved. However when someone claims that he is fighting for the honor of Hashem he will always claim that although he personally would concede ground to the other side but he has no right to compromise on Hashem’s “Cheshbon” and therefore the argument will never be resolved. The lesson to be learned is, don’t turn personal disagreements into religious battles.