Do scriptures allow Hindus to marry foreigners, or non-Hindus of other races?
Inter-caste marriage has its restrictions, but what about interracial marriages?
No, the Dharma Shastras explicitly forbid Hindus from marrying foreigners, otherwise known as Mlecchas.
Here are some verses from the Atri Smriti that prohibit interracial marriages:
The purification for knowing [having sex with] a Mleccha woman consists in a Santapanam penance, or it is said, one may attain to purification by performing a Taptakrcchra penance.
If a person lives with a wife known [who has had sex with] by a Mleccha man, he becomes purified by bathing with the cloth on and drinking clarified butter.
By cohabiting [living with and having sex with], unknowingly, with the women of the Chandalas, Mlecchas ... one is purified by a Paraka.
But if one knows them willingly, and procreates children with them, then he is degraded to the same caste as them; there is no doubt about it, for that man is born as her son.
So it is actually a huge sin to marry foreigners, since one becomes degraded to the caste of Mleccha, and there seems to be no expiation for knowingly having sex with a Mleccha.
This answer explains some of the different types of penances.