Short answer
No one is sent to Hell for a Muslim's sins. A hadith in Sahih Muslim (2767) does speak of Jews and Christians and the Fire (Hell). It says that on the Day of Judgement God (﷾) gives each Muslim a Jew or a Christian as his release from the Fire. Another version says God (﷾) forgives some Muslims' sins, as heavy as mountains, and lays them on the Jews and Christians. A hadith in Sahih al-Bukhari gives the key. Everyone who enters Paradise is shown the place he would have had in the Fire. The one who dies rejecting faith enters the Fire for his own unbelief and sins. The scholar al-Nawawi says the disbeliever takes the place in the Fire that a forgiven believer would have had. The sins laid on him are like those sins, and they are his own. So the Qur'an's rule stands: no one carries another's sin. The hadith speaks of God's (﷾) judgement on the Last Day, and God (﷾) alone knows who dies rejecting faith.
Full explanation
The hadith critics quote
Christian critics quote three versions of one hadith in Sahih Muslim. A hadith is a report of what the Prophet (ﷺ) said. The critics say Islam also punishes one person in another's place. They say Muslims object to exactly this in the Christian teaching about the cross. They add that it breaks the Qur'an's own rule.
Abu Musa reports that the Prophet (ﷺ) said that on the Day of Resurrection Allah will hand every Muslim a Jew or a Christian and say: this is your release from the Fire.
Abu Burda reports from his father that the Prophet (ﷺ) said no Muslim dies without Allah putting a Jew or a Christian into the Fire in his place.
Abu Burda reports from his father that the Prophet (ﷺ) said some Muslims will come on the Day of Resurrection with sins as heavy as mountains; Allah will forgive them those sins and place them on the Jews and the Christians (Arabic: wa yada'uha 'ala al-yahud wa al-nasara). A narrator adds "as far as I reckon", showing some doubt about the wording.
Everyone has two places
The Prophet (ﷺ) taught that everyone who enters Paradise is shown the place he would have had in the Fire had he rejected faith. Everyone who enters the Fire is shown the place he would have had in Paradise. The believer is shown the place in the Fire he escaped. The disbeliever is shown the place in Paradise he lost.
Abu Hurayra reports that the Prophet (ﷺ) said everyone who enters Paradise will be shown the place he would have had in the Fire had he rejected faith, so that he is more thankful; and everyone who enters the Fire will be shown the place he would have had in Paradise had he believed, so that he feels the loss.
What al-Nawawi says about it
Al-Nawawi (d. 1277) is the best-known scholar to have explained Sahih Muslim line by line. He explains the hadith by the two places. The disbelievers enter the Fire for their own unbelief and their own sins. So the believers are spared the places that were theirs. On the version with sins like mountains, he says God (﷾) forgives the Muslims those sins and removes them. The words "places them on" are not literal. Those who die rejecting faith carry sins like those, and the sins are their own. He reads it this way because of the Qur'an's rule.
Al-Nawawi explains "your release from the Fire" by the hadith that every person has a place in Paradise and a place in the Fire: Allah has ordained a number to fill the Fire, and when the disbelievers enter it for their own disbelief and sins, the Muslims are released. On the version with sins like mountains, he says Allah forgives the Muslims those sins and removes them, and places the like of them on the Jews and Christians because of their own disbelief and sins, so they enter the Fire for their own deeds and not for the Muslims' sins. He says the words "places them on" are figurative: Allah places on them sins like those, which are their own. He reads it this way because the Qur'an says no bearer of burdens bears another's burden.
No one carries another's sin
The Qur'an states the rule in several places. Every soul earns only for itself. No bearer of burdens bears another's burden, not even for a close relative. God (﷾) also punishes no one until a messenger has reached them.
“Say: "Shall I seek for (my) Cherisher other than Allah, when He is the Cherisher of all things (that exist)? Every soul draws the meed of its acts on none but itself: no bearer of burdens can bear the burden of another. Your goal in the end is towards Allah: He will tell you the truth of the things wherein ye disputed."”
“Nor can a bearer of burdens bear another's burdens if one heavily laden should call another to (bear) his load. Not the least portion of it can be carried (by the other). Even though he be nearly related. Thou canst but admonish such as fear their Lord unseen and establish regular Prayer. And whoever purifies himself does so for the benefit of his own soul; and the destination (of all) is to Allah.”
“Namely, that no bearer of burdens can bear the burden of another; That man can have nothing but what he strives for;”
“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).”
Why this is not one person paying for another
Christians teach that Jesus died for sinners, so that an innocent man suffers for the guilty. Nothing like that happens in this hadith. No innocent person is punished. No one is punished for a Muslim's sin. The believer is forgiven by God's (﷾) mercy. The disbeliever answers for his own unbelief and his own deeds. The Bible's prophet Ezekiel gives the same rule: the soul that sins is the one that dies.
Abu Burda reports from his father that the Prophet (ﷺ) said some Muslims will come on the Day of Resurrection with sins as heavy as mountains; Allah will forgive them those sins and place them on the Jews and the Christians (Arabic: wa yada'uha 'ala al-yahud wa al-nasara). A narrator adds "as far as I reckon", showing some doubt about the wording.
“The soul that sinneth, it shall die. The son shall not bear the iniquity of the father, neither shall the father bear the iniquity of the son: the righteousness of the righteous shall be upon him, and the wickedness of the wicked shall be upon him.”
The hadith, version by version
Sahih Muslim 2767a — Kitab al-Tawba (The Book of Repentance), hadith 2767a
Abu Musa reports that the Prophet (ﷺ) said that on the Day of Resurrection Allah will hand every Muslim a Jew or a Christian and say: this is your release from the Fire.
Cited for
Each Muslim is handed a Jew or a Christian as his "release from the Fire", or one is put into the Fire in his place. So a non-Muslim pays the price that saves the Muslim.
The response
The Prophet (ﷺ) taught that each person is shown the place he would have had in the other abode. Al-Nawawi explains that a believer's place in the Fire is left empty. The disbeliever enters the Fire for his own unbelief and takes that place. He pays for nothing but his own deeds.
Where it lands — The "release" is the believer's escape from a place that is then filled by someone who earned it himself.
Sahih Muslim 2767d — Kitab al-Tawba (The Book of Repentance), hadith 2767d
Abu Burda reports from his father that the Prophet (ﷺ) said some Muslims will come on the Day of Resurrection with sins as heavy as mountains; Allah will forgive them those sins and place them on the Jews and the Christians (Arabic: wa yada'uha 'ala al-yahud wa al-nasara). A narrator adds "as far as I reckon", showing some doubt about the wording.
Cited for
Muslims' sins as heavy as mountains are forgiven and laid on the Jews and Christians. So they are punished for sins that are not theirs, which is exactly what the Qur'an forbids.
The response
Al-Nawawi explains that God (﷾) forgives the Muslims those sins and removes them. He says the words "places them on" mean that God (﷾) places sins like them on the Jews and Christians. They carry sins like those, but the sins are their own, earned by their own unbelief and deeds. He reads it this way because the Qur'an says no one bears another's burden.
Where it lands — No one is punished for a Muslim's sin, so the hadith agrees with the Qur'an's rule.
Perspectives
The critics' reading
Christian critics say this hadith gives Islam its own kind of substitution. They say Jews and Christians go to the Fire so that Muslims can be spared. They say this contradicts the Qur'an's rule that no one bears another's burden.
One version of Sahih Muslim 2767 says God will forgive some Muslims' sins, as heavy as mountains, and lay them on the Jews and Christians. Critics read this as punishing one person for another's sins.
Rebuttal
The hadith does speak of sins placed on Jews and Christians. Al-Nawawi, writing in the 1200s, explains that God (﷾) places on them sins like those, earned by their own unbelief and deeds. In the Christian teaching of the cross an innocent man is punished for the guilty. Here no innocent person is punished, and no one bears another's burden (35:18).
The Islamic perspective
The hadith describes an exchange of places, not of guilt. The Prophet (ﷺ) taught that each person is shown the place he would have had in the other abode. Al-Nawawi explains that the disbeliever enters the Fire for his own unbelief and sins. He fills the place the forgiven believer would have had.
Sahih al-Bukhari 6569 says that everyone who enters Paradise is shown the place he would have had in the Fire. It says everyone who enters the Fire is shown the place he would have had in Paradise.
Al-Nawawi explains Sahih Muslim 2767 by the two places. He says the Jews and Christians enter the Fire for their own unbelief and sins, not for the sins of the Muslims.
The Qur'an teaches that no bearer of burdens bears another's burden. It teaches that each soul earns only for itself.
In the hadith no innocent person is punished. So it has nothing in common with an innocent man punished for the guilty.