What should one who falsely swore on the Qur'an do out of fear of his wife being divorced?

The basic ruling is that it is forbidden to swear falsely; this is the al-Yamin al-Ghamus (the plunging oath) and is considered one of the major sins. If you were forced to swear falsely, and the wife's family was capable of causing a separation, then it is hoped that there will be no sin upon you. However, if you were not coerced, then you have erred, and you must repent sincerely, feel remorse, and it is more cautious to offer expiation for the oath.