What is the ruling on managing the money of an addicted brother who does not handle his finances wisely, and is it obligatory to inform him of his share in his mother's inheritance if that will lead to him selling his share and squandering the money?

Spending money on sinful acts is folly and lack of sound judgment. A person who is foolish (safeeh) may be interdicted, but this is the prerogative of the judge, not individuals. Your action of taking your brother's money and giving him what he needs might be in his interest, but it carries the risk of his money being lost or inherited by those who are not his rightful heirs. Therefore, his case must be brought before the judiciary to prove his folly, so that the judge may appoint a guardian to safeguard his money.