He also called for an end to the “cancer of corruption”, saying it took money away from development .
Mr Obama made the comments in the first ever address by a US leader to the 54-member AU at its headquarters in the Ethiopian capital, Addis Ababa.
The address marked the climax of Mr Obama’s five-day trip to Africa.
He visited Kenya and Ethiopia, the headquarters of the AU.
African leaders should respect their constitutions and step down when their term ends.
Violence in Burundi following President Pierre Nkurunziza’s bid for a third term showed how stability could be threatened if constitutional rules were ignored, he said.
“Nobody should be president for life,” Mr Obama said.
“I don’t understand why people want to stay so long, especially when they have got a lot of money,” he added.
Democracy existed in name but not in substance when journalists were jailed and activists were threatened, he said.
Corruption was “draining billions of dollars” from Africa, he added.
The money could be used to build schools and hospitals, Mr Obama said.
The rapid economic growth in Africa was changing “old stereotypes” of a continent hit by war and poverty, he said.
But unemployment needed to be urgently tackled on a continent whose one-billion people will double in a few decades, Mr Obama said.
“We need only look to the Middle East and North Africa to see that large numbers of young people with no jobs and stifled voices can fuel instability and disorder,” he added. –BBC