Marriages between Muslims and non Muslims is prohibited under Islamic law, except in certain limited circumstances. Malaysian Islamic law is even more limited than some other schools of Islamic law. International human rights norms however suggest that the prohibition and criminalisation of interfaith marriages is an unreasonable restriction on the rights of men and women to marry and found a family.
Under Islamic law, as I understand it, a Muslim woman can only marry a Muslim man. However, a Muslim man can marry a Muslim woman or a Kitabiyyah. See for example section 10 of the Islamic Family Law (Federal Territories) Act 1984:-
Section 10. Persons of other religions.
(1) No man shall marry a non-Muslim except a Kitabiyah.
(2) No woman shall marry a non-Muslim.
A Kitabiyyah is generally regarded as a Person of the Book i.e. a Jew or a Christian. Traditional commentators discourage marriages between Muslims and non Muslims, although there is a divergence of views on when it is allowed and what restrictions are placed on the general rule above. See for example Abdur Rahman I’ Doi.
Most Malaysian State Syariah legislation, following the Shafie school, puts a slight gloss on the above rule. The following is the definition of a Kitabiyyah taken from the FT legislation:-
Kitabiyah means -
(a) a woman whose ancestors were from the Bani Ya’qub; or
(b) a Christian woman whose ancestors were Christians before the prophethood of the Prophet Muhammad; or
(c) a Jewess whose ancestors were Jews before the prophethood of the Prophet ‘Isa;
*Prophet ‘Isa is of course Christ to Christians. As I understand it, Bani Ya’qub are the descendants of the Prophet Ya’qub (Jacob To the Christians?), who had 12 sons who formed the 12 tribes of Israel.
(Incidentally, this means only Malayalee Christians in Malaysia – the Mar Thomites and the Syrian Christians – would probably comply with this requirement).
Some States do not have this provision at all, instead only allowing Muslims to marry Muslims. (I recently learnt that Sarawak only permits a Muslim to marry another Muslim, whether you are a man or a woman.)
An interesting discussion on this, which also shows this Malaysian definition of a Kitabiyyah is perhaps unique to the Shafie school can be seen in the English Court of Appeal decision in U Viswalingam v S Viswalingam  1 MLJ 10.
Malaysian civil law
The Malaysian Law Reform (Marriage and Divorce) Act 1976 does not apply to Muslims (section 3). Hence, in Malaysia there is no means for a Muslim to marry a non Muslim under the civil law. In fact, the Muslim who goes through a marriage or even cohabits with a non Muslim runs the risk of falling foul of a number of offences under State Syariah criminal law prohibiting fornication (zina) and close proximity (khalwat).
This position is of course contrasted with the position in India and Singapore where by the Special Marriage Act of 1958 and the Woman’s Charter respectively, a Muslim and a non Muslim can lawfully marry. In both countries, if two Muslims are getting married to each other, their marriage is solemnised under Islamic law. It is noteworthy that the famous Bollywood actor Shah Rukh Khan, a Muslim, and his Hindu wife Gauri have been happily married since 1991.
International human rights norms
In addition to the protections for religious freedom and equality under the law, there are specific protections for the human right, in of itself, to found a family.
For example, the Universal Declaration of Human Rights provides that “Men and women of full age, without any limitation due to race, nationality or religion, have the right to marry and to found a family. They are entitled to equal rights as to marriage, during marriage and at its dissolution.”: Article 16(1)
Article 23(2) of the International Covenant on Civil and Political Rights provides that “The right of men and women of marriageable age to marry and found a family shall be recognized”. Although the Human Rights Committee of the United Nations in General Comment No. 19 does not explicitly state that this provision requires the State to ensure the right to persons of differing religions to marry, it does state that “the right to freedom of thought, conscience and religion implies that the legislation of each State should provide for the possibility of both religious and civil marriages.”
It is also interesting that in the USA, the equality clause of the US Bill of Rights has been held to prohibit the criminalisation of marriages between difference races: see the US Supreme Court decision of Richard Perry Loving et. ux v Virginia (1967) 388 US 1, 18 L Ed 2d 1010 (now dramatised in the Hallmark production “Mr & Mrs Loving”). There does not seem to be any case on interfaith marriages. However, the current dispute in the US on the legality of same gender marriages and how it is ultimately resolved will no doubt be of relevance to this issue.
It seems clear, however, based on a simple reading of the provisions of the UDHR and ICCPR that to restrict the right of two consenting adults to live together and to found a family merely because of their differing religions is contrary to the rights of equality under the law, the right of religious freedom and the right to found a family contained within international human rights norms.