Refernce(s) for prohibitting it
Those who may say that it is not allowed for Muslims to read the Torah and the Bible or any other "Holy Book" would refer to a narration which was compiled by al-Bazzar and Imam Ahmad and qualified as weak by some scholars:
It was narrated from Jaabir ibn ‘Abdullah (may Allah be pleased with him) that ‘Umar ibn al-Khattaab (may Allah be pleased with him) came to the Prophet (blessings and peace of Allah be upon him) with some written material he had got from one of the people of the Book. He read it to the Prophet (blessings and peace of Allah be upon him), and he got angry and said: Are you confused (about your religion), O son of al-Khattaab? By the One in Whose hand is my soul, I have brought it (the message of Islam) to you clear and pure. Do not ask them about anything, lest they tell you something true and you disbelieve it, or they tell you something false and you believe it. By the One in Whose hand is my soul, if Moosa were alive, he would have no option but to follow me.”.
On this basis one could prohibit reading any other holy book except the Quran as it should be enough for us and as it states that it is the most (or absolutely) correct revelation. One could also add this statement of ibn 'Abbas from sahih al-Bukhari:
"Why do you ask the people of the scripture about anything while your Book which has been revealed to Allah's Messenger (ﷺ) is newer and the latest? You read it pure, undistorted and unchanged, and Allah has told you that the people of the scripture changed their scripture and distorted it, and wrote the scripture with their own hands and said, 'It is from Allah,' to sell it for a little gain. Does not the knowledge which has come to you prevent you from asking them about anything? No, by Allah, we have never seen any man from them asking you regarding what has been revealed to you!"
which goes ahead with (29:51). See also this hadith from sahih al-Bukahri which is quote saying that some of the people of the book used to translate their scripture to Muslims.
That's why scholars like ..
Al-Qurtubi (may Allah have mercy on him) said:
If the one who reads it – namely the Qur’an – will have a tenfold reward or more for each letter, according to what we mentioned in the introduction to this book, then turning away from it and towards other Books is misguidance and loss, and it is a bad deal and waste of time. (Fatwa)
But one can also say: At the time this happened our religion Islam was still not fully established, as some rules may have come later. For example many people say that the Prophet prohibited writing down or compiling his hadith which seems to be a similar case (Be aware that there are narrations showing that our Prophet even allowed it to some of his companions). So if we assume both cases are similar than it after the last revelation of the Quran it should have been allowed to cross-check the truth of Quran with other earlier revelations. Maybe it was at the beginning frowned about because there were still pagans, and people were still close to worshiping idols, so maybe they could have a fall back! This means one could say the Question is: should we understand this Prohibition for 'Omar as a general prohibition for all Muslims or was it due to the circumstances a more likely temporary prohibition?
Most scholars agree that only a well-versed with strong faith person (a scholar for instance) is allowed to read those books to be able to debate non-Muslims and establish proofs against their false teachings!
Al-Haafiz Ibn Hajar (may Allah have mercy on him) said (in fath al-Barry):
With regard to this matter, it is important to note that in the case of those who are not well-versed in knowledge and are lacking in faith, it is not permissible for them to read any of those books. (From the same fatwa as above)
this means for ordinary people the above prohibition applies:
An-Nawawi (may Allah have mercy on him) said (in rawdat at-Talibeen):
The books of the Torah and Gospel are among the things that it is prohibited to seek benefit therein, because they changed and altered them.
A-Rahibani (May Allah have mercy on him) said (in matalib uwli aN-nuha):
It is not permissible to study the books of the people of the Book, based on textual evidence, because the Prophet (blessings and peace of Allah be upon him) got angry when he saw ‘Umar (may Allah be pleased with him) carrying a page of the Torah. … And they should not read books that contain both truth and falsehood, or narrate what they learn from them, because that may be detrimental to belief.
(From the same fatwa as above)
So the major opinion among scholars seem discouraging reading books of other faiths and beliefs. I think it depends on the faith of the person and the intention, so one couldn't say it's generally prohibited or allowed.
For more details read the full linked fatwa and see also this fatwa in Arabic. They show that even if the majority of scholars discourage reading Bible and Torah there's a strong enough evidence contradicting their verdict. Of course the major reason that may defend their view to some extent is the fact that they say it is prohibited for the ordinary Muslim (only).
Reference for a permission
Those who may say that it is allowed for Muslims to read the Torah and the Bible or any other "Holy Book" would refer to this Hadith which was compiled by Imam Ahmad in his Musnad and abu Dawod in his sunan:
relate traditions from the children of Isra'il; there is no harm.
A longer versions can be found in sahih al-Bukhari and jami' at-Tirmdihi both on the authorithy of Abdullah ibn 'Amr.
In sahih Muslim you may find a hadith which doesn't refer to bani Israel (the children of Israel or Jews) at all and has apparently a similar wording it is one of the evidences for not allowing to take down hadith.
The (shared and displayed) statement of the hadith was interpreted as follows: It is allowed to narrate traditions of bani Israel which are in their books (without checking the narrator chains).
Al-Khattabi added that this doesn't allow to tell lies about them.
Imam Malik said: this means that it is allowed to
narrate about them what is good. While itz is not allowed to tell one knows is a lie.
Imam a-Shafi'i said: it means narrate about them what you are sure that it is not a lie, and what you can tolerate of it there's no harm in narrating it. And he quoted a statement of a hadith as an evidence:
... Whatever the people of the Book tell you, do not verify them, nor falsify them ...
and there was neither a prohibition nor a permission to tell what one is absolutely certain about.
a-Tybi explained: There's no contradiction between the permission in this hadith and the prohibition in an other neither for narrating from or about nor reading in their books, because here he meant telling their stories like killing themselves for repentance. While the prohibition was about following their rulings because they have been made void by his shari'a or at the beginnings of Islam when the rulings were not settled yet, so afterwards he gave permission as the reason for the prohibition was no more given.
These statements are taken and translated (by myself) from this Arabic fatwa.
Some works that have discussed other faiths based on a deep insight of their texture and rulings
AFAIK only a few scholars wrote books and discussed other faith based on a deep insight of the rulings and theology of these religions. For example ibn Hazm the Andalusian (dhahiri scholar) in his Al-Fasl fi al-Milal wa al-Nihal الفصل في الملل والأهواء والنحل (The Separator Concerning Religions, Heresies, and Sects) and later al-Shahrastani the Persian (sunni shafi'i scholar) in his Kitāb al–Milal wa al-Nihal كتاب الملل والنحل (The Book of Sects and Creeds) these are the major works in this subject. While most Muslim scholars like al-Ghazaly have primarily discussed the false believes and thoughts of sects which have been cerated within Islam or based on Islamic traditions. Ibn Hazm of course had the opportunity to grow up in a more liberal Islamic environment in which Jews and Christians played a big role however his book also covers other religious faiths like Hindu etc.
