Short answer
No, and the New Testament itself says so. Many Protestant Christians teach "once saved, always saved". They mean that a true believer can never be lost, because Christ paid for every sin in advance, even sins not yet done. They add that a true believer will not go on sinning. But the letter to the Hebrews warns believers themselves. If we sin wilfully after knowing the truth, no sacrifice for sins remains, only a fearful wait for judgement. Jesus said that not everyone who calls him Lord will enter heaven, but the one who does his Father's will. So what a believer does still matters for his end. Islam agrees that deeds count. Salvation comes by God's (﷾) mercy to those who believe and do good deeds, and God (﷾) forgives again and again those who repent.
Full explanation
What Christians teach
Christians hold that the death of Christ was one offering, made once for all. They say it perfects those who are made holy. God (﷾) promises to remember their sins no more. Paul, a Christian missionary of the first century, says salvation is by grace through faith, not by works. He says whoever confesses Jesus as Lord and believes God (﷾) raised him will be saved. Christians add that this is no licence to sin. Paul answers "God forbid", and says believers are created for good works. Many Protestants go further and teach eternal security. They say a true believer can never be lost, because every sin, even one not yet done, was paid for on the cross.
“For by one offering he hath perfected for ever them that are sanctified.”
“And their sins and iniquities will I remember no more.”
“For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: Not of works, lest any man should boast.”
“That if thou shalt confess with thy mouth the Lord Jesus, and shalt believe in thine heart that God hath raised him from the dead, thou shalt be saved.”
“What shall we say then? Shall we continue in sin, that grace may abound? God forbid. How shall we, that are dead to sin, live any longer therein?”
“For we are his workmanship, created in Christ Jesus unto good works, which God hath before ordained that we should walk in them.”
No sacrifice left for wilful sin
The same chapter of Hebrews that speaks of one offering for ever also warns the believer. Whoever sins wilfully after receiving the knowledge of the truth has no sacrifice left, only judgement. The writer then compares this with the man who despised the Law of Moses and died without mercy. So the offering does not cover a believer whatever he later does. His end still depends on how he lives after he believes.
“For if we sin wilfully after that we have received the knowledge of the truth, there remaineth no more sacrifice for sins, But a certain fearful looking for of judgment and fiery indignation, which shall devour the adversaries.”
“He that despised Moses’ law died without mercy under two or three witnesses: Of how much sorer punishment, suppose ye, shall he be thought worthy, who hath trodden under foot the Son of God, and hath counted the blood of the covenant, wherewith he was sanctified, an unholy thing, and hath done despite unto the Spirit of grace?”
Good works are still required
Hebrews tells believers to provoke one another to love and good works. James, another New Testament letter, asks what use faith is without works. It answers that a man is justified by works and not by faith only. Ezekiel, a prophet of the Old Testament, had already said that a righteous man who turns to sin will die in it. His past righteousness will not be remembered.
“And let us consider one another to provoke unto love and to good works:”
“What doth it profit, my brethren, though a man say he hath faith, and have not works? can faith save him?”
“Ye see then how that by works a man is justified, and not by faith only.”
“But when the righteous turneth away from his righteousness, and committeth iniquity, and doeth according to all the abominations that the wicked man doeth, shall he live? All his righteousness that he hath done shall not be mentioned: in his trespass that he hath trespassed, and in his sin that he hath sinned, in them shall he die.”
What Jesus said
Jesus said that calling him "Lord, Lord" is not enough. Only the one who does the will of his Father enters the kingdom. A man asked him how to gain eternal life, and he answered: keep the commandments. He warned that whoever breaks the least commandment and teaches others to do so will be least in the kingdom. None of this fits a promise that belief alone settles a man's future sins in advance.
“Not every one that saith unto me, Lord, Lord, shall enter into the kingdom of heaven; but he that doeth the will of my Father which is in heaven.”
“And he said unto him, Why callest thou me good? there is none good but one, that is, God: but if thou wilt enter into life, keep the commandments.”
“Whosoever therefore shall break one of these least commandments, and shall teach men so, he shall be called the least in the kingdom of heaven: but whosoever shall do and teach them, the same shall be called great in the kingdom of heaven.”
Life after believing is still a test
Believers still live for years after they first believe. The Bible's own answer to why is that life is still a trial. James says the crown of life comes to the man who endures temptation, when he has been tried. A life still under trial is a life whose end is not yet settled. The Qur'an says the same. Allah (﷾) created death and life to test which of you is best in deed. People will not be left alone on saying "We believe" without being tested.
“Blessed is the man that endureth temptation: for when he is tried, he shall receive the crown of life, which the Lord hath promised to them that love him.”
“He Who created Death and Life, that He may try which of you is best in deed: and He is the Exalted in Might, Oft-Forgiving;-”
“Do men think that they will be left alone on saying, "We believe", and that they will not be tested?”
The good man who never heard of Christ
Take a man who gives his life to charity and good works. Through no fault of his own, he never hears of Christ. Where is the justice for him? Many Christians read John 14:6 and Acts 4:12 to mean there is no salvation without faith in Jesus. This is the view of many evangelicals, Christians who stress personal faith in Christ. On that reading, such a man is lost whatever he did. For centuries whole peoples lived and died without hearing the name. The Qur'an teaches that Allah (﷾) does not punish until He has sent a messenger to warn. It also teaches that no one bears another's burden.
“Jesus saith unto him, I am the way, the truth, and the life: no man cometh unto the Father, but by me.”
“Neither is there salvation in any other: for there is none other name under heaven given among men, whereby we must be saved.”
“Who receiveth guidance, receiveth it for his own benefit: who goeth astray doth so to his own loss: No bearer of burdens can bear the burden of another: nor would We visit with Our Wrath until We had sent a messenger (to give warning).”
What Islam teaches
The Qur'an ties salvation to faith and righteous deeds together. It tells both Muslims and the People of the Book (Jews and Christians) that their wishes will not prevail. Whoever does evil will be repaid for it. No one carries another's burden, and each person has only what he strives for. The door of repentance stays open. Allah (﷾) forgives again and again those who repent, believe and do right. He can turn their evil deeds into good. Forgiveness does not end the worship of the forgiven. The Prophet (ﷺ) prayed at night for so long that his feet cracked. Aisha asked why he did it, since Allah (﷾) had forgiven his faults of the past and those to follow. He answered that he should love to be a thankful slave of Allah (﷾). That promise came from God (﷾) to His Prophet (ﷺ), by revelation. It was not a payment that covers every believer's future sins.
“Not your desires, nor those of the People of the Book (can prevail): whoever works evil, will be requited accordingly. Nor will he find, besides Allah, any protector or helper.”
“If any do deeds of righteousness,- be they male or female - and have faith, they will enter Heaven, and not the least injustice will be done to them.”
“By (the Token of) Time (through the ages), Verily Man is in loss, Except such as have Faith, and do righteous deeds, and (join together) in the mutual teaching of Truth, and of Patience and Constancy.”
“Namely, that no bearer of burdens can bear the burden of another; That man can have nothing but what he strives for;”
“"But, without doubt, I am (also) He that forgives again and again, to those who repent, believe, and do right, who,- in fine, are ready to receive true guidance."”
“Unless he repents, believes, and works righteous deeds, for Allah will change the evil of such persons into good, and Allah is Oft-Forgiving, Most Merciful,”
“Say: "O my Servants who have transgressed against their souls! Despair not of the Mercy of Allah: for Allah forgives all sins: for He is Oft-Forgiving, Most Merciful.”
Aisha reports that the Prophet (ﷺ) prayed at night for so long that his feet cracked. She asked why he did it, since Allah had forgiven his faults of the past and those to follow. He answered: should he not love to be a thankful slave of Allah?
Verse by verse
Hebrews 10:17 — Epistle to the Hebrews, chapter 10, verse 17
“And their sins and iniquities will I remember no more.”
Cited for
God has promised to remember the believer's sins no more, so every sin, past and future, is already forgiven in Christ.
The response
Nine verses later the same writer says that for wilful sin after knowing the truth there is no more sacrifice, only judgement. The promise in verse 17 cannot mean what the later verses deny.
Where it lands — Hebrews does not teach that believers are forgiven in advance whatever they do.
Hebrews 10:26-27 — Epistle to the Hebrews, chapter 10, verses 26-27
“For if we sin wilfully after that we have received the knowledge of the truth, there remaineth no more sacrifice for sins, But a certain fearful looking for of judgment and fiery indignation, which shall devour the adversaries.”
Cited for
Many Christians read "sin wilfully" as turning away from Christ altogether, not the everyday sins of a believer. They note that verse 29 speaks of treading underfoot the Son of God.
The response
Even on that reading, the warning is written to "we", the believers. It ends in fiery judgement for the one who turns away. The writer does not treat the cross as having settled a man's end, whatever he does after he believes. Some reply that those who fall away were never true believers. But the writer says "we", and speaks of the blood "wherewith he was sanctified" (10:29). He treats them as sanctified believers.
Where it lands — Hebrews warns believers that what they do after believing matters to their end, as Jesus and the Qur'an teach.
Ephesians 2:8-9 — Epistle to the Ephesians, chapter 2, verses 8-9
“For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: Not of works, lest any man should boast.”
Cited for
Salvation is by grace through faith, not of works, so deeds play no part in it.
The response
The next verse says believers are created unto good works. James asks whether faith without works can save, and answers that a man is justified by works and not by faith only. Jesus said the doer of the Father's will enters the kingdom.
Where it lands — The New Testament as a whole will not let faith stand apart from deeds.
Perspectives
Mainstream Christian
Christ's one offering saves believers by grace through faith. Good works follow true faith, but they do not earn salvation, and grace is no excuse to keep sinning.
Hebrews says Christ's single offering has perfected for ever those who are sanctified (made holy).
Rebuttal
Hebrews does say this. But later in the same chapter, Hebrews warns that for wilful sin after knowing the truth no sacrifice is left, only judgement. So "for ever" cannot mean that a believer's end is settled whatever he does.
Paul teaches that salvation is by grace through faith and not of works.
Rebuttal
Paul does say this. But James says a man is justified by works and not by faith only. And Jesus made doing his Father's will the way into the kingdom.
Many Protestants hold that the cross covers a true believer's sins, past and future. God, they say, remembers them no more.
Rebuttal
Hebrews 10:17 does say God (﷾) will remember their sins no more. But nine verses later the same writer says that for wilful sin there is no more sacrifice, only judgement. Ezekiel too says a righteous man who turns to sin will die in it.
The Islamic perspective
Salvation is by God's (﷾) mercy to those who believe and do good. No sacrifice covers sins in advance, and no one can carry another's sins. Anyone who repents is forgiven.
Hebrews says that for wilful sin after knowing the truth no sacrifice remains, only judgement.
Paul too says grace is no licence to sin.
Jesus said entry to the kingdom depends on doing the Father's will and keeping the commandments.
Some teach that any believer's future sins are already paid for, whatever he does. That removes the fear of judgement that the Bible uses to restrain him.
James says the crown of life is given to the one who endures when he is tried. So the believer's life after he believes is still a test.
A teaching that faith in Jesus alone saves leaves no justice for a righteous man who never heard of him. The Qur'an says Allah (﷾) punishes no one until a messenger has warned them.
The Qur'an teaches that a person is judged by faith and deeds, not by wishes. Allah (﷾) forgives those who repent.