Does the son bear the burden of sin if his mother spends the money he gives her on unlawful things or on things that are of no benefit, even though he knows of her mismanagement, and he does so to avoid disobedience?

If the mother is provided for by the father, the son is not obligated to spend on her.

If the mother is extravagant, this is a form of foolishness that permits interdiction over her. It is not permissible for the son to give her money while she is in this state, due to Allah’s saying: "And do not give the foolish [disposed] your property, which Allah has made a means of sustenance for you." The son's refusal to give her money in this case is not disobedience; rather, it is a form of dutifulness to her by preventing her from falling into disobedience of Allah.