Is informing the boss about a grave mistake committed by a colleague who works with him in the same institution considered a sin against the colleague, and what should be done regarding the colleague's refusal to forgive the informer?

If your neighbor's mistake did not result in harm to you or the institution, then reporting him is considered backbiting (namimah), and you should have warned him and covered for him, especially if reporting him would harm him, for a neighbor has a greater right to be covered for.

However, if covering for him would result in harm to you or the institution, then you must report it out of trustworthiness and sincere advice. There is no sin upon you for this, and your neighbor must forgive you.

You should reconcile with your neighbor and appease him, for the rights of a neighbor are great.