The above answer gives some good points, but it does not give a cause or reason. The sons of Eli were wicked, but if we follow the narrative of 1 Samuel and more importantly, the book of Judges, then we could arrive at a better answer.
One of the darkest periods of Israel's history was during the time of the Judges. There is a continued cycle of sin, disobedience, idolatry and rebellion. It starts after the death of Joshua, who did not raise up a godly leader to lead the people after his death. By this time the priesthood is in shambles! For this reason, the book of Judges has a stock sentence:
"People did what was right in their own eyes." Judges 17:6, 21:25.
When we move into the time of 1 Samuel, we are still very much in the period of the judges, as Samuel is the last judge, before we move into the time of the kings, of whom Saul is first. It is within this context of immorality that we read of Eli's sons. They offered wicked sacrifices, slept with women at the temple, and could care less about God. They are a product of the time. In addition, Eli is no model father. He generally looked the other way. All of this is to show that this was a dark period in the life of Israel. What they needed was a king. David would be this man.
The answer to this can be found in I Samuel, Chapter 2. The basic answer is that they did not follow in their father's footsteps because they were wicked men.
Eli's sons were actually priests since Eli's whole line had been made priests by God. But Eli's sons used their position to abuse others. They would take the best meats from the sacrifices for their own food. They would use their power to get to sleep with women who served in religious positions.
Because of their actions, a prophet came and prophesied that they would not cointinue to be priests and that the two sons of Eli would, in fact, both die on the same day.
Most likely Eli was lazy and didn't bother to punish his sons for bad behavior. He might not have even taken time to reward them for good behavior either. Being in charge of children is very hard work at times. You can't let kids get their way when you say "Don't!" or not punish them when you said you would. The only way to keep kids in line other than upholding the rules 99 percent of the time is to lash out at them like a rabid animal and strike fear into them. Rewards should also be given for good behavior.
They were wicked man and Eli did not hold the reigns of his household.