Short answer
Yes, for the living. You can make dua for a non-Muslim's guidance, health, safety, success and wellbeing. After they die, you may not ask Allah to forgive them or have mercy on them, but you can speak well of them and comfort their family. All four schools agree.
Dua for a living non-Muslim's guidance and worldly good is permitted and praiseworthy. Asking forgiveness for a non-Muslim who has died is haram, and Hanafi texts warn that it is very grave. Asking forgiveness for a living non-Muslim is understood as asking that Allah guide them to Islam so that they are forgiven.
The four schools agree on this.
Explanation
The Prophet ﷺ prayed for non-Muslims many times. When the tribe of Daws refused Islam and his companions asked him to curse them, he prayed instead: O Allah, guide Daws. Abu Hurayrah asked him to pray for his mother, who was not yet Muslim, and she accepted Islam the same day. When Jews sneezed near him hoping he would say "may Allah have mercy on you", he said "may Allah guide you and set your affairs right".
- For the living: pray for guidance, health, protection, a good job, healing from illness, safety on a journey. "O Allah, guide my father and open his heart to Islam" is one of the best duas a convert can make.
- When they sneeze: say yahdikumullah wa yuslih balakum (may Allah guide you and set your affairs right).
- After death: the Quran forbids asking forgiveness for those who died on shirk, even close family. Abraham stopped praying for his father's forgiveness once it became clear.
What can you say when a non-Muslim dies? Words of comfort and true praise: "He was a generous man", "I am sorry for your loss". Avoid "may God have mercy on him" or "rest in peace" said as a prayer. This is not coldness. Their affair is with Allah, the Most Just and Most Merciful, and we do not speak about anyone's final state. The four schools agree on this.
For duas you can use for sick people, see dua for visiting the sick and dua for parents.
The evidence
Quran, Surah At-Tawba 9:113
مَا كَانَ لِلنَّبِىِّ وَٱلَّذِينَ ءَامَنُوٓا۟ أَن يَسْتَغْفِرُوا۟ لِلْمُشْرِكِينَ وَلَوْ كَانُوٓا۟ أُو۟لِى قُرْبَىٰ مِنۢ بَعْدِ مَا تَبَيَّنَ لَهُمْ أَنَّهُمْ أَصْحَٰبُ ٱلْجَحِيمِ ﴿١١٣﴾
It is not for the Prophet and those who have believed to ask forgiveness for the polytheists, even if they were relatives, after it has become clear to them that they are companions of Hellfire.
Asking forgiveness for those who died on shirk is not allowed.
Quran, Surah At-Tawba 9:114
وَمَا كَانَ ٱسْتِغْفَارُ إِبْرَٰهِيمَ لِأَبِيهِ إِلَّا عَن مَّوْعِدَةٍۢ وَعَدَهَآ إِيَّاهُ فَلَمَّا تَبَيَّنَ لَهُۥٓ أَنَّهُۥ عَدُوٌّۭ لِّلَّهِ تَبَرَّأَ مِنْهُ ۚ إِنَّ إِبْرَٰهِيمَ لَأَوَّٰهٌ حَلِيمٌۭ ﴿١١٤﴾
And the request of forgiveness of Abraham for his father was only because of a promise he had made to him. But when it became apparent to Abraham that his father was an enemy to Allah, he disassociated himself from him. Indeed was Abraham compassionate and patient.
Abraham prayed for his father while alive, then stopped.
Sahih al-Bukhari 2937
Sahih... اللهم اهد دوسا وأت بهم ...
"O Allah, guide Daws and bring them."
The Prophet ﷺ prayed for the guidance of a tribe that had refused Islam.
Sahih Muslim 2491
Sahih... اللهم اهد أم أبي هريرة ...
"O Allah, guide the mother of Abu Hurayrah."
Praying for a non-Muslim parent's guidance.
Sunan Abi Dawud 5038
Graded sahih... يهديكم الله ويصلح بالكم ...
"May Allah guide you and set your affairs right."
The reply to a non-Muslim who sneezes.
Where the ruling is recorded
- Radd al-Muhtar (Ibn Abidin), Kitab al-Salah, the discussion of dua at the end of the prayer. Dua for forgiveness for a deceased disbeliever is forbidden; dua for a living one's guidance is allowed.
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.