Is it obligatory to inform the employer about the actions of a colleague who conducts personal business and interests outside of work and neglects his duties, or should one remain silent and bear the responsibility? And does silence entail sin?

You should advise your colleague who is negligent in his work and threaten him with informing the employer. If he does not respond, then it is obligatory upon you to inform the employer, because it is not permissible to cover for him and deceive the employer, for "religion is sincerity (or advice)." Ibn Hazm mentioned that none of the people of Islam say it is permissible to cover for a Muslim who has wronged another Muslim. Al-Khadimi mentioned that if concealing a matter results in financial or physical harm to someone, or violates a legal ruling, then you must inform and make it known if it is unknown, and testify if requested. Otherwise, you should conceal it.