Is it permissible to continue using a Mus'haf that my father took from their neighbor, who left the village due to war, fearing that it might be touched by someone who desecrates Mus'hafs?

If the owner of the-Mushaf is known, you should contact him to seek permission and inform him that the-Mushaf is in your possession. There is no harm in continuing to use it, as it is highly probable that he would not object to it. Some scholars have explicitly permitted benefiting from books without the owners' permission, while others have forbidden or disliked it. The hadith prohibiting looking into another's book without their permission is weak and is interpreted as referring to a book containing a secret or a trust. As for books of knowledge, it has been said that it is permissible to look into them without permission because knowledge should not be withheld. However, it has also been said that it is not permissible due to the apparent meaning of the hadith.