Is it considered a sin not to respond to the requests of a brother who is mentally and physically ill, who makes excessive trivial demands and relies on others, causing exhaustion and anger due to his affairs?

It is recommended to be kind to the sick and considerate of their feelings. Whoever can benefit their brother should do so, for the most beloved of people to Allah are those most beneficial to people, and among the most beloved deeds is bringing joy to a Muslim and relieving their distress. If the patient's requests are difficult, one can gently apologize without angering them or hurting their feelings, as Allah Almighty says: "And if you turn away from them, seeking mercy from your Lord which you hope for, then speak to them a gentle word." There is no sin in not responding to unnecessary requests from the patient, while being careful not to rebuke them, as Allah Almighty says: "And as for the beggar, do not chide him."