Short answer
With respect, and with the Qur'an as the measure. A Muslim believes in the Torah God (﷾) gave Moses and the Injil God (﷾) gave Jesus (the Gospel he preached). The Qur'an never tells him to follow the Bible as it now stands. It calls itself a guardian over the earlier books, and it says men changed some of their words. So a Muslim tests each passage against the Qur'an. Where the Bible teaches one God (﷾), the prophets, the moral law and the judgement, he sees the old revelation still speaking. Where it contradicts the Qur'an or itself, or charges a prophet with grave sins, he sees the work of men. The Prophet (ﷺ) told Muslims neither to believe nor to deny what the People of the Book (Jews and Christians) read to them. Scholars apply this where the Qur'an is silent. A Muslim need not defend every line of it.
Full explanation
The Qur'an stands guard over the earlier books
The Qur'an describes itself with the Arabic word muhaymin (a guardian and witness) over the scripture before it. It confirms what is true there and is the judge of the rest. The command to Muslims themselves is to follow what their Lord has sent down to them.
“To thee We sent the Scripture in truth, confirming the scripture that came before it, and guarding it in safety: so judge between them by what Allah hath revealed, and follow not their vain desires, diverging from the Truth that hath come to thee. To each among you have we prescribed a law and an open way. If Allah had so willed, He would have made you a single people, but (His plan is) to test you in what He hath given you: so strive as in a race in all virtues. The goal of you all is to Allah; it is He that will show you the truth of the matters in which ye dispute;”
“Follow (O men!) the revelation given unto you from your Lord, and follow not, as friends or protectors, other than Him. Little it is ye remember of admonition.”
The Qur'an says men changed the text
The Qur'an does not send Muslims to the Bible as it now stands. It speaks of men who wrote the Book with their own hands and said it was from God (﷾). It speaks of some who moved words out of their places or twisted the Book with their tongues. It says Christians forgot a good part of the message. The Bible itself carries a similar warning through Jeremiah, a prophet of Israel. He speaks against the scribes' pen and against those who perverted the words of the living God (﷾). The Qur'an says the Torah was entrusted to its rabbis and doctors of law to protect. The only Book it says God (﷾) Himself will guard is the Qur'an.
“Then woe to those who write the Book with their own hands, and then say:"This is from Allah," to traffic with it for miserable price!- Woe to them for what their hands do write, and for the gain they make thereby.”
“There is among them a section who distort the Book with their tongues: (As they read) you would think it is a part of the Book, but it is no part of the Book; and they say, "That is from Allah," but it is not from Allah: It is they who tell a lie against Allah, and (well) they know it!”
“Of the Jews there are those who displace words from their (right) places, and say: "We hear and we disobey"; and "Hear what is not Heard"; and "Ra'ina"; with a twist of their tongues and a slander to Faith. If only they had said: "What hear and we obey"; and "Do hear"; and "Do look at us"; it would have been better for them, and more proper; but Allah hath cursed them for their Unbelief; and but few of them will believe.”
“From those, too, who call themselves Christians, We did take a covenant, but they forgot a good part of the message that was sent them: so we estranged them, with enmity and hatred between the one and the other, to the day of judgment. And soon will Allah show them what it is they have done.”
““ ‘How do you say, “We are wise, and Yahweh’s law is with us?” But, behold, the false pen of the scribes has made that a lie.”
“And the burden of the Lord shall ye mention no more: for every man’s word shall be his burden; for ye have perverted the words of the living God, of the Lord of hosts our God.”
“It was We who revealed the law (to Moses): therein was guidance and light. By its standard have been judged the Jews, by the prophets who bowed (as in Islam) to Allah's will, by the rabbis and the doctors of law: for to them was entrusted the protection of Allah's book, and they were witnesses thereto: therefore fear not men, but fear me, and sell not my signs for a miserable price. If any do fail to judge by (the light of) what Allah hath revealed, they are (no better than) Unbelievers.”
“We have, without doubt, sent down the Message; and We will assuredly guard it (from corruption).”
Agrees, contradicts, or is silent
A Muslim sorts what he reads into three kinds. Some passages agree with the Qur'an. Moses says, "Hear, O Israel: The Lord our God is one Lord". Mark has Jesus repeat it as the first of all the commandments. There a Muslim hears the old revelation still speaking. Some passages contradict the Qur'an. Exodus says Aaron made the golden calf. The Qur'an says a man called the Samiri led the people astray while Aaron warned them. There a Muslim follows the Qur'an and sees the hand of men in the other account. On much else, such as most of the Bible's family lists and the details of its histories, the Qur'an is silent. Those passages he neither believes nor denies. Christians often quote 5:47, 5:68, 10:94 and 6:115 to send Muslims to the Bible. Those verses are answered on the pages "Does the Qur'an tell Muslims to follow the Bible?" and "Does the 'Islamic dilemma' prove the Qur'an wrong?".
“Hear, O Israel: The Lord our God is one Lord:”
“And Jesus answered him, The first of all the commandments is, Hear, O Israel; The Lord our God is one Lord:”
“And he received them at their hand, and fashioned it with a graving tool, after he had made it a molten calf: and they said, These be thy gods, O Israel, which brought thee up out of the land of Egypt.”
“(Allah) said: "We have tested thy people in thy absence: the Samiri has led them astray."”
“Aaron had already, before this said to them: "O my people! ye are being tested in this: for verily your Lord is (Allah) Most Gracious; so follow me and obey my command."”
A Muslim believes in the Torah God (﷾) gave Moses and the Injil God (﷾) gave Jesus (the Gospel he preached).
Must he follow the Bible as it now stands?
No. The Qur'an never tells him to, and it says men changed some of the words.
What is his measure?
The Qur'an. It calls itself muhaymin, a guardian and witness over the books before it.
Does the passage agree with the Qur'an?
Then he hears the old revelation still speaking, as in “Hear, O Israel: The Lord our God is one Lord”.
Does it contradict the Qur'an, or itself?
Then he sees the work of men. Exodus says Aaron made the golden calf. The Qur'an says the Samiri led the people astray while Aaron warned them.
Is the Qur'an silent?
Then he neither believes nor denies it, as the Prophet (ﷺ) taught.
Ask for the text, then weigh it
Where the Qur'an speaks to the People of the Book about their scripture, it asks them to bring it and read it out. That is how one asks for evidence. It tells the people of the Injil to judge by what God (﷾) revealed in it. That means what God (﷾) revealed, wherever it survives, not every line now printed in the Bible. The Prophet (ﷺ) told Muslims neither to believe nor to deny what the People of the Book (Jews and Christians) read to them. They were to say they believe in God (﷾) and in what He sent down. Muslim scholars hold that this covers reports the Qur'an neither confirms nor denies. The Prophet (ﷺ) also allowed Muslims to relate the stories of the Children of Israel.
“All food was lawful to the Children of Israel, except what Israel Made unlawful for itself, before the Law (of Moses) was revealed. Say: "Bring ye the Law and study it, if ye be men of truth."”
“Let the people of the Gospel judge by what Allah hath revealed therein. If any do fail to judge by (the light of) what Allah hath revealed, they are (no better than) those who rebel.”
Abu Hurayra reports that the People of the Book used to read the Torah in Hebrew and explain it in Arabic to the Muslims. The Prophet (ﷺ) told the Muslims neither to believe them nor to call them liars, but to say: we believe in Allah and in what was revealed to us (Qur'an 2:136).
Abdullah ibn Amr reports that the Prophet (ﷺ) said: pass on what you have from me, even a single verse; you may relate the reports of the Children of Israel, and there is no harm in it; and whoever lies about me on purpose will take his seat in the Fire.
Perspectives
The Islamic perspective
A Muslim believes in the revelations given to Moses and Jesus, and reads today's Bible with respect and open eyes. The Qur'an, the last Book and the one God (﷾) kept, is the measure. What agrees with it is the old revelation still speaking. What contradicts it is the hand of men. The Prophet (ﷺ) taught Muslims to leave the rest unconfirmed and undenied.
The Qur'an calls itself a guardian and witness over the scripture before it.
The Qur'an tells people to follow the revelation sent down to them from their Lord.
The Qur'an says some of the People of the Book wrote scripture with their own hands and moved words out of their places. It says Christians forgot a good part of the message.
Moses and Jesus both taught that the Lord our God (﷾) is one Lord, as the Qur'an teaches.
Exodus says Aaron made the golden calf. The Qur'an says the Samiri led the people astray and Aaron warned them.
The Prophet (ﷺ) told Muslims neither to believe nor to deny what the People of the Book read to them.
The Prophet (ﷺ) allowed Muslims to relate the stories of the Children of Israel.
Where the Bible agrees with the Qur'an, a Muslim recognises the original revelation surviving in it. Where it contradicts the Qur'an or itself, he takes it as the work of men.