Ruling on swearing on the Mushaf and the expiation for breaking such an oath

Firstly: 

An oath is not binding
unless it is sworn by one of the names of Allaah or by one of His
attributes. 

Swearing on the Qur’aan is
swearing by the word of Allaah which is one of His attributes. As for
swearing by the Mushaf, if what is meant is the words of Allaah contained
therein, then it is an Islamically acceptable oath, but if what is meant is
the paper and ink, then it is an oath by something other than Allah, which
is shirk, as the Prophet (peace and blessings of Allaah be upon him)
said: “Whoever swears by something other than Allaah has committed an act of
kufr or shirk.” Narrated by al-Tirmidhi (1535) and Abu Dawood (3251);
classed as saheeh by al-Albaani in Saheeh al-Tirmidhi. 

Hence it is better not to
swear by the Mushaf, because the Mushaf contains the words of Allaah, and it
contains paper and ink. 

As for putting one’s hand
on the Mushaf or inside it, this is an innovated thing which some people do
for emphasis to warn the one who is swearing the oath against lying. 

Shaykh Ibn ‘Uthaymeen (may
Allaah have mercy on him) was asked about the ruling on swearing by the
Mushaf.

 He (may Allaah have mercy
on him) replied: It is not permissible to make vows or swear oaths except by
Allaah or by one of His attributes. If a person swears by Allaah, may He be
glorified and exalted, then there is no need to bring the Mushaf for him to
swear on it, because swearing on the Mushaf was not done at the time of the
Prophet (peace and blessings of Allaah be upon him) or at the time of
the Sahaabah; even after the Mushaf was compiled in book form they did not
swear on it, rather a person would swear by Allaah without swearing on the
Mushaf. End quote from Fataawa Noor ‘ala al-Darb. 

It says in a statement of
the Islamic Fiqh Council: 

2 – It is not essential for
the person who is swearing the oath to place his hand on the Mushaf, Torah
or Gospel when swearing for his oath to be valid, but it is permissible if
the judge deems it essential so as to warn the one who is swearing the oath
against lying. 

3 – It is not permissible
for the Muslim to place his hand on the Torah or Gospel when swearing an
oath, because these copies that are in circulation are distorted and that
are not in the original form that was revealed to Moosa and ‘Eesa (peace be
upon them), and because the sharee’ah with which Allaah sent His Prophet
Muhammad (peace and blessings of Allaah be upon him) has abrogated the
laws that came before it. 

End quote from Fataawa
Islamiyyah (3/463) 

Secondly: 

If a person swears an oath
then he cannot fulfil that oath, he has to offer expiation. The expiation
for breaking an oath (kafaarat yameen) is explained in the answer to
question no. 45676. 

And Allaah knows best.