Short answer
No. In Sunni belief, no one who dies with even a grain of faith stays in Hell forever. Some sinful Muslims may be punished for a time, unless Allah forgives them, and then they are brought out and admitted to Jannah. Hell is everlasting for those who die in disbelief.
Believers who commit major sins and die without repenting are under Allah's will: He may forgive them, or punish them for a time and then admit them to Jannah. Those who die upon disbelief remain in the Fire forever. Jannah and Jahannam are both created now and will never end.
The four schools agree on this.
Explanation
The Prophet ﷺ said that whoever said "There is no god but Allah" with a barley grain's weight of good in his heart will be brought out of the Fire, then whoever had a wheat grain's weight, then an atom's weight. This hadith in al-Bukhari, with many others like it, is the basis of the Sunni belief that faith, however small, ends in Jannah.
Intercession plays a part. The Prophet ﷺ said his intercession is for the people of major sins in his ummah. Angels, prophets and believers will also intercede by Allah's permission, and in the end Allah brings out by His own mercy people who did no good beyond their faith. See intercession.
For those who die rejecting faith, the Quran is clear that they will abide in Hell eternally (98:6), and Allah does not forgive shirk for one who dies upon it (4:48). Some of the early Muslim sects, the Kharijites and the Mu'tazila, held that a major sinner stays in Hell forever. Sunni scholars rejected this because of the many authentic texts above.
This is not a reason to take sins lightly. Even a short time in Hell is beyond anything a person can bear, and no one knows whether Allah will forgive him without punishment. Every person will pass over the sirat above Hell (19:71-72), and Allah will save those who feared Him. The right attitude is hope and fear together: repent, do good and trust in His mercy. Read more in Jannah in Islam.
The evidence
Quran, Surah An-Nisaa 4:48
إِنَّ ٱللَّهَ لَا يَغْفِرُ أَن يُشْرَكَ بِهِۦ وَيَغْفِرُ مَا دُونَ ذَٰلِكَ لِمَن يَشَآءُ ۚ وَمَن يُشْرِكْ بِٱللَّهِ فَقَدِ ٱفْتَرَىٰٓ إِثْمًا عَظِيمًا ﴿٤٨﴾
Indeed, Allah does not forgive association with Him, but He forgives what is less than that for whom He wills. And he who associates others with Allah has certainly fabricated a tremendous sin.
Sins below shirk are under Allah's will to forgive.
Quran, Surah Al-Bayyina 98:6
إِنَّ ٱلَّذِينَ كَفَرُوا۟ مِنْ أَهْلِ ٱلْكِتَٰبِ وَٱلْمُشْرِكِينَ فِى نَارِ جَهَنَّمَ خَٰلِدِينَ فِيهَآ ۚ أُو۟لَٰٓئِكَ هُمْ شَرُّ ٱلْبَرِيَّةِ ﴿٦﴾
Indeed, they who disbelieved among the People of the Scripture and the polytheists will be in the fire of Hell, abiding eternally therein. Those are the worst of creatures.
Eternal punishment is for those who die in disbelief.
Quran, Surah Maryam 19:71-72
وَإِن مِّنكُمْ إِلَّا وَارِدُهَا ۚ كَانَ عَلَىٰ رَبِّكَ حَتْمًۭا مَّقْضِيًّۭا ﴿٧١﴾ ثُمَّ نُنَجِّى ٱلَّذِينَ ٱتَّقَوا۟ وَّنَذَرُ ٱلظَّٰلِمِينَ فِيهَا جِثِيًّۭا ﴿٧٢﴾
And there is none of you except he will come to it. This is upon your Lord an inevitability decreed. Then We will save those who feared Allah and leave the wrongdoers within it, on their knees.
All pass over Hell, and the God-fearing are saved.
Sahih al-Bukhari 44
Sahih... يَخْرُجُ مِنَ النَّارِ مَنْ قَالَ لاَ إِلَهَ إِلاَّ اللَّهُ وَفِي قَلْبِهِ وَزْنُ شَعِيرَةٍ ...
"Whoever said there is no god but Allah and has in his heart a barley grain's weight of good will come out of the Fire."
Believers will not stay in Hell forever.
Sunan Abi Dawud 4739
Graded sahih... شَفَاعَتِي لأَهْلِ الْكَبَائِرِ مِنْ أُمَّتِي ...
"My intercession is for the people of major sins in my ummah."
The Prophet will intercede for sinful believers.
Where the ruling is recorded
- al-Aqida al-Tahawiyya, Section on the major sinners of the ummah. States that major sinners of the ummah will not stay in Hell forever if they died as believers.
- Sharh al-Aqa'id al-Nasafiyya (al-Taftazani), Section on major sins. Refutes the view that a major sinner stays in Hell forever.
This answer is pending review by the scholars on our team. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.