Is it permissible to marry off a sister who is mentally ill to a poor, married man who is older than her, especially with suspicions about his coveting her money, and would her guardian be sinful if he does not marry her to him?

If the suitor is pleasing in his religion and character, then accept him, for the Messenger of Allah, peace and blessings be upon him, said: "If there comes to you one whose religion and character you approve, then marry him. If you do not do so, there will be tribulation on earth and widespread corruption." It is not permissible for you to prevent your sister from marrying, as this is a form of 'adl (unjust prevention of marriage) which is prohibited. The husband is obliged to provide for his wife's maintenance and accommodation. The wife and her guardian have the right to seek divorce if the husband is unable to fulfill this. If you suspect that the man is coveting your sister's wealth, and your sister is not prudent in managing her money, you can place her under interdiction by referring the matter to a judge to appoint you as a guardian over her funds.