Is it permissible for the husband to consummate his marriage with his wife, to whom he is already religiously wedded after giving the Mahr, when her father refuses to allow him to do so unless he furnishes three complete rooms, which he is unable to do?

You are not permitted to consummate the marriage with your wife except with the knowledge of her father and her family. You must comply with what her father has stated, or refer the matter to the Sharia court for arbitration. Do not violate this by secluding yourselves with her alone, as this leads to evil, harm, and discord. You must be patient and resolve issues through understanding with her father, or by involving mediators. If that is not possible, then your matter should be referred to the Sharia court.