Is the refusal of the questioner to drive his mother to her sister—because she will smoke there—considered an act of disobedience (ʿuqūq), and is he obligated to obey her in this matter?

It is not permissible to abandon one's parents due to their disobedience; rather, one must treat them with kindness. If a mother asks her son to take her to visit her sister, and this is for the purpose of maintaining kinship ties, then it is permissible for the son to take her, and there is no sin upon him, even if there is a possibility of a sin occurring there, as long as the trip itself is not for the purpose of committing that sin. This distinction is legally recognized between one who disobeys by his travel and one who disobeys during his travel.

However, if it is highly probable that the mother is going there to smoke cigarettes, then taking her would be assisting in a sin, so do not obey her. Apologize gently and kindly, and advise her. But if you know that she will go with a non-mahram (definition) driver if you do not take her, then you should take her, in order to minimize the harm, as her going with a non-mahram is a greater harm. This might also make her more receptive to your advice.