Is it permissible, in Islamic law, to refuse financial assistance to one's brother, knowing that he is unemployed and his wife is the sole provider, and that his request for money burdens his sister, who lives alone and has a limited income, and he creates problems when refused?

If your brother is in need of financial support, and you possess more than your own needs, then you must spend on him appropriately. However, if your earnings only suffice for your own needs and there is nothing extra, then you are not obligated to do so. For the obligation of maintenance to apply, the recipient of the maintenance must be poor, the provider must have surplus wealth or earnings, and the recipient must not be capable of earning a livelihood.