What is the ruling on the testimony of a non-believer, with details, may Allah reward you with goodness?

The fundamental principle is that a witness must be a just Muslim, based on Allah's saying: ﴿وَأَشْهِدُوا ذَوَيْ عَدْلٍ مِنْكُمْ﴾ (And call to witness two just persons from among you) and ﴿إِنْ جَاءَكُمْ فَاسِقٌ بِنَبَأٍ فَتَبَيَّنُوا﴾ (If a mischievous person comes to you with any news, ascertain the truth). The testimony of a male or female betrayer, or a male or female fornicator, is not permissible.

The testimony of disbelievers is not accepted unless no other option is available. Abu Hanifa, Ahmad, Shurayh, Ibrahim al-Nakha'i, and al-Awza'i made an exception for the testimony of a disbeliever regarding a Muslim's will during travel out of necessity, based on Allah's saying: ﴿يَا أَيُّهَا الَّذِينَ آمَنُوا شَهَادَةُ بَيْنِكُمْ إِذَا حَضَرَ أَحَدَكُمُ الْمَوْتُ حِينَ الْوَصِيَّةِ اثْنَانِ ذَوَا عَدْلٍ مِنْكُمْ أَوْ آخَرَانِ مِنْ غَيْرِكُمْ﴾ (O you who have believed, testimony between you when death approaches one of you at the time of bequest is [that of] two just men from among you or two others from outside [your community]).

As for the testimony of a disbeliever against a disbeliever, Malik, Shafi'i, and Ahmad prohibited it, while Abu Hanifa permitted it, citing the stoning of two Jews based on the testimony of other Jews against them for adultery. Al-Sha'bi, Ibn Abi Layla, and Ishaq permitted the testimony of a Jew against a Jew, but prohibited the testimony of a Jew against a Christian or a Zoroastrian.