What does it mean that a story is an Israelite narration? Is there a Hadeeth (narration) that belongs to Israelite narrations and should we act according to it?

All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.

The term "Israelite narrations" was taken from "Israel" the name of Prophet Ya‘qoob (Jacob) . The term extensively refers to the narrations that Muslims quoted from the People of the Book, Jews and Christians, orally or through their writings. Muslims reported these narrations from them to learn lessons and take admonition, based on the saying of the Prophet, sallallaahu ‘alayhi wa sallam, that was narrated by 'Abdullaah ibn 'Amr :"Relate traditions from the children of Israel, and you bear no sin." [Al-Bukhari]

This applies to the narrations that are in accordance with the Quran and Sunnah (Prophetic tradition). Otherwise, it is not permissible to relate them or learn lessons from them. The author of ‘Awn Al-Ma‘bood Sharh Sunan Abu Daawood said that Maalik said, "The Hadeeth means that it is permissible to relate the traditions that are good, and as for what is known to be a lie then no (it is not permissible)."

The Hadeeths of the Prophet, sallallaahu ‘alayhi wa sallam, are not considered Israelite narrations -- even those texts that talk about the Children of Israel. They are Islamic reports, and what was authentically narrated from the Prophet, sallallaahu ‘alayhi wa sallam, is the truth. This is because Allaah The Almighty Says about the Prophet, sallallaahu ‘alayhi wa sallam, (what means): {Nor does he speak from [his own] inclination. It is not but a revelation revealed.} [Quran 53:3-4]

Allaah Knows best.