The BBC's Rachel Harvey, who is at the scene, says many people, particularly young men, are trying to return to their homes to check on their livestock and property.
Checkpoints, manned by police and volunteers, have been set up on main roads into the exclusion zone.
However, there are lots of small lanes that give access to the slopes, and this is making it very difficult for the authorities to keep track of people, our correspondent says.
"We'll do everything we can to stop them," Hadi Purnomo, the district chief in Sleman, told the Associated Press.
He described several nearby villages as "death zones", saying: "There's no life there. The trees, farms, houses are scorched. Everything is covered in heavy gray ash."
"People are going home with sacks of grass to feed their cows. Some say they need to see the condition of their houses while others want to pick up belongings as they don't have enough clothes at the shelters," checkpoint volunteer Wawan Fauzi told AFP.
Many people ignored the initial warnings of an impending eruption.
Map showing location of Mount Merapi volcano
At least 29 people are now confirmed dead, according to the Disaster Management Agency.
A two-month-old baby was among the dead.
The man known as the volcano's spiritual gatekeeper, Mbah or grandfather Marijan, was also found dead in his house about 4km (2.5 miles) from the summit.
