I remember reading long ago that, according to Jewish law, you may not identify as a convert a person whom you know is a convert. Is this correct and what is the reference?
For whether it is permitted to ask, see this question.
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There is a clear prohibition to hurt others, and particularly a convert, with words. As the Rambam writes (MT Hilchot Mechirah 14:12-13)
Just as the prohibition against ona'ah applies with regard to business transactions, it applies with regard to speech, as Leviticus 25:17 states: "A person should not abuse his colleague, and you shall fear your God. I am the Lord"; this refers to verbal abuse.
What is implied? If a person is one who has repented, one should not say: "Remember your initial deeds." If a person is a descendant of converts, one should not tell him: "Remember your ancestors' deeds."
Therefore, to the extent that referring to a convert as such would be hurtful to him, it is clearly prohibited. Should you know for sure that it isn't hurtful to him then it would not be prohibited.
Mentioning it to others would fall within the purview of lashon hara and is just as forbidden. The Chofetz Chaim writes (Prohibition Against Lashon Hara 4:1)
It is forbidden to speak against one's friend — even if not to his face and even if it be true — something that will shame him. And not only demeaning things in general, such as mentioning about him the [negative] deeds of his fathers and his relatives, or his early deeds, both those between him and his Maker and [those between] him and his neighbour; for since he now conducts himself correctly it is forbidden to demean him with this and it is called lashon hara.