Praise be to Allah.
It is not permissible for a Muslim to swear by anything except Allaah, may He be exalted, alone, because the Prophet (peace and blessings of Allaah be upon him) said: “Whoever swears an oath, let him swear by Allaah, or else keep silent.”
It is not essential for a person swearing an oath to put his hand on the Mus-haf (copy of the Qur’an) for the oath to be valid, but some people do this to make the oath more binding, so that the person swearing the oath will be afraid to tell lies.
It is not permissible for the Muslim to put his hand on the Tawraat or Injeel when swearing an oath, because the copies in circulation are distorted, and are not the original Books that were revealed to Moosa and ‘Eesaa (peace be upon them); and because the Message with which Allaah sent His Prophet Muhammad (peace and blessings of Allaah be upon him) abrogates the messages that came before.
If the system in a country that is not ruled by Islamic law requires the person swearing an oath to put his hand on the Tawraat or Injeel, or both, then the Muslim should ask the court to let him swear by Allaah, may He be exalted. There is nothing wrong with him putting his hand on the Holy Qur’aan, which is the Word of Allaah and is therefore one of His attributes. If they do not allow him this request, and he is forced to swear by the Tawraat or Injeel, then this is considered to be compulsion, and there is no sin on him if he puts his hand on one or both of them, so long as he does not intend to show respect to them by doing so. And Allaah knows best.