Hating people is prohibited; fighting or feuding is even more strongly prohibited. For a feud is a deep-seated, long-running dispute which leads to multiple confrontations and draws more and more people into the vortex of hatred. When the animus is directed against a righteous Torah scholar, it is even worse, because it causes the public to turn away from Torah and mitzvot. Worst of all is when a student feuds with his or her teacher. This is why it was so extremely serious when Koraĥ (and his followers) fought against Moses, teacher of all Israel, and humblest of men.
Sometimes people think they are helping their friends or family by joining their vendetta against someone. However, they are really hurting them because they are worsening the feud and fanning the flames of hatred. Real friends try to save their friends from quarrels that will just be painful and sap their energy.
At the same time, there is value in a disagreement focused on actual ideas, which is not motivated by hatred, and is carried out respectfully. While it can still cause tension and hurt feelings, it does help clarify issues. This type of disagreement is called a “disagreement for the sake of heaven,” and it endures because its purpose is to clarify the truth. Ultimately it will become clear there was some truth on both sides. This is why the Talmudic debates of Shammai and Hillel have staying power. However, when a disagreement is motivated by personal grudges and hatred, neither side has value. Hence it will ultimately not prevail. Before it fizzles out though, it will sap the energies of all those involved.
