In the ancient world there were two main types of fuel - dung and olive oil. Dung is a by-product of the animal and is primarily used for cooking - olive oil is a by-product of the olive and is primarily used for lighting a lamp. With these two fuels in mind the writer of Kings draws a striking picture that illuminates God's feelings toward the lines of David and Jeroboam (son of Nebat).
The dung line -
“I (Yahwh) will cut off from Jeroboam every male, both bond and free in Israel, and will burn up the house of Jeroboam, as a man burns up dung until it is all gone.” (1 Kings 14:10 ESV)
Unrelenting syncretism (mixing Yahwism with Canaanite idolatry) and immediate destruction of Jeroboam's line (death of his son - Nadab - 1 Kings 15:25-27).
The olive oil line -
“Nevertheless, for David’s sake the LORD his God gave him a lamp in Jerusalem, setting up his son after him, and establishing Jerusalem.” (1 Kings 15:4 ESV)
Continuation of Davidic line until 586 BCE (destruction of Judah) - resulting in the first return of the King (Matthew 1) and to be culminated in the final return of the warrior king - Jesus (Zechariah 14:4-5).