Why is an intimate friendship with opposite gender not permissible? Where does Allah command this?

It is impermissible for the very fact that it produces illegitimate children, fornication and adultery. An intimate friend in one whom you share your thoughts with and are very close, you tell everything you can to this person.

In Islamic system we know that this is not allowed. Allah’s messenger (ﷺ) said,

“When a man and a woman are alone, the third person is the devil.” [1]

Narrated `Uqba bin 'Amir: Allah's Messenger (ﷺ) said,

"Beware of entering upon the ladies." A man from the Ansar said, "Allah's Apostle! What about Al-Hamu the in-laws of the wife (the brothers of her husband or his nephews etc.)?" The Prophet (ﷺ) replied: The in-laws of the wife are death itself." [2]

 Islam has segregated men and women in societies, in mosques, in families etc. This is clear evidence, hence a man and woman cannot be intimate friends. Narrated Umar ibn al-Khattab (رضى الله عنه), Tirmidhi transmitted it. The Prophet (ﷺ) said,

"Whenever a man is alone with a woman the Devil makes a third." [3]

References:

[1] [Narrated by Ahmad, al-Tirmidhi and al-Haakim; classed as saheeh by al-Albaani in Saheeh al-Jaami’ (2546)]

[2] [Sahih al-Bukhari 5232, Book 67, Hadith 165]

[3] [Ahmad and Al-Tirmidhi 3118]