"Kol Yisrael areyvim zeh lazeh" "All Israel are responsible for one another" (see Talmud Sanhedrin 27b; Shavuos 39a)
This talmudic dictum is the source that we are responsible as Jews to make sure (to the best of our ability) that our fellow Jews also keep Torah and mitzvos.
Rashi in his commentary to Deut. 29:28, explains that once Israel crossed the Jordan and accepted the oath on Mt. Gerizim and Mt. Eval, they became one community and obligated (for all time) to make sure that none of their fellows would sin, or else be punished themselves for lack of taking responsibility for their neighbors actions.
So, if your fellow Jew sins, and you could have prevented it, you are also liable to punishment, and are obliged to try and stop the other person from sinning.
Levit. 19:17 states: ("Hocheyach Tochiyach") "You shall not hate your brother in your heart. You shall surely rebuke your fellow, but you shall not bear a sin on his account."
From here we learn the general obligation that every Jew is obligated to rebuke another who is sinning.
Levit. 19:14 also tells us: "lifnei iver" "...You shall not place a stumbling block before a blind person..."
It is a mitzvah to not cause another Jew to violate a Torah prohibition. (see Rambam Sefer HaMitzvos negative command 299)
Based on these verses and chazal, it is instantly understood that one must help a Jew who does not know they are Jewish by letting him/her know and showing them how to practice Judaism. Certainly, one should not be a party to allowing a Jewish child to be given to a gentile family.
A "tinok shenishbah" or an "infant that was captured" by gentiles, is the subject of Talmud shabbos 68b. There, Rav and Shmuel both agree that although the child will spend many Sabbaths violating the laws of Shabbos unknowingly and unwittingly, the child will eventually be obligated to bring a korban (sin offering) as soon as they find out they are Jewish and what Shabbos really means. (even Rabbi Yochanan and Reish Lakish agree that the child is living a life of sin albeit forcibly, so as not to require the official offering) The Rambam paskens (decides) according to Rav and Shmuel, that the Halachah requires such a person to make up for their sins with a korban. (see Rambam Hilchos Shegagos 2:6 ; 7:2)
Therefore, we certainly have an obligation to help such a "tinok shenishbah" become aware they are Jewish, so they can stop sinning and keep Shabbos (as well as all the Torah).
The Rambam says that anyone who does not believe in Torah sheh'baal peh, is not even considered in the category of "Israel". However, if a child of such a person grew up in that father's community, (of heretics) we should treat that child as a "tinok shenishbah" and try to speak with that child in a peaceful manner in order to bring them back to Judaism.
(see Rambam Hilchos Mamrim 3:2-3 )
It should be learned from this Rambam, that a Jewish child living among gentiles (who are certainly not in the category of "Israel") would also be treated like a "tinok shenishbah". We would have to inform the child he is Jewish and re-affiliate them in a friendly way.
Gemara Gittin 47a:
"If someone sells himself and his children to idol-worshippers, we do not redeem him but we redeem the children."
"(We redeem the children) because of the ruin (that may befall them), how much more so, where there is a chance of murder."
From here we see that even if there is no risk of murder, we would redeem children (even without their parents) back into the Jewish community.
Gemara Gittin 57a:
The Rabbis taught: It once happened that Rabbi Yehoshua the son of Chananya was travelling to a great city in Rome. They said to him, "There is a baby in the prison who has beautiful eyes and a fine appearance, and his locks are arranged in curls"....
The story continues that R' Yehoshua found out he was a Jewish child and redeemed him. The child grew up to be R' Yishmael ben Elisha.
R Eliezer Silver - Vaad Hatzalah story:
In 1945 R' Eliezer Silver (eventually chief Rabbi of the United States and Canada) and Dayan Grunfeld of London, both went on a mission to post war Europe. They were tasked with finding lost Jewish children in orphanges. They came to a monastary in France which housed an orphanage for victims of the war years. R' Silver asked that the Jewish children be handed over. The priests denied there were any Jewish children there. So, R' Silver went to the children by bedtime and started reciting the bedtime Shma Yisroel verses out loud. Suddenly, many children in the crowd started crying for their parents and answering the shma. R' Silver liberated those children.
The story is mentioned in a recent post by Aish.com here: