If I steal some money or property from a muslim brother and I don't apologize or give it back, I have to give my good deeds or take his sins on the day of judgement. But what if I cheat/steal/damage the property of a non muslim and do'nt give it back? what will happen on the day of judgement ? As the non muslim has no hope for paradise and destined for hell unless Allah wish. Jazakallah khairan.

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

It is established that every wronged person will have retribution exacted against those who wronged him on the Day of Judgment, whether he is a Muslim or non-Muslim. It was narrated in Musnad Ahmad on the authority of ‘Abdullaah ibn Unays, may Allaah be pleased with him, that the Prophet said: “‘(Allah Almighty will say on the Day of Judgment) … No one of the people of Hell who has a grievance against any of the people of Paradise shall enter Hell until I settle that grievance. And no one of the people of Paradise whom any of the people of Hell has a grievance against shall enter Paradise until I settle that grievance. This will take place for all grievances, even a slap on the face.’ We asked: ‘How will that be done when we will be resurrected before Allah naked, uncircumcised, and having nothing with us?’ He said: ‘By means of exchange of good deeds with bad deeds.’”

Therefore, whoever does injustice to a Muslim and does not seek his pardon (clear his liability from such a grievance) in this worldly life, the manner of exacting retribution against him was reported in the Sunnah. It was narrated on the authority of Abu Hurayrah in Saheeh Al-Bukhari that the Messenger of Allah said: “Whoever has wronged his brother should ask for his pardon, as (in the Hereafter) there will be neither a dinar nor a dirham, (he should secure pardon in this worldly life) before some of his good deeds are taken and credited to his brother, or, if he has done no good deeds, some of the bad deeds of his brother are taken and cast on him.”

If the wronged person is a non-belligerent non-Muslim, then – as far as we know – there is no reported text underlining the manner of such exacting of retribution through the exchange of good deeds and bad deeds.

Allah Knows best.