The authorities fixed Monday's deadline in response to deadly clashes in the city that has left dozens of people dead.
The authorities said anyone left inside the protest zone would be considered "a criminal" and would face up to two years in prison.
It offered to provide buses to take the protesters home if they left voluntarily.
The government's move is aimed at choking off the red shirts, who have occupied a roughly 3sq km protest zone barricaded by fuel-soaked tyres and bamboo spikes in Bangkok's shopping and tourist hub since early April.
Our correspondents said the protesters do not trust the government and fear they will be arrested if they get on the buses.
There are also fears that the move to get the women, children and elderly to leave is a precursor to an all-out assault on the protest site.
Earlier on Monday, the chief military strategist of the red shirts died in a Bangkok hospital.
Major-General Khattiya Sawasdipol, better known as Seh Daeng or Commander Red, had been accused by the government of being a renegade soldier and creating a paramilitary force for the so-called red shirt protesters.
His death, reported by the Vajira hospital where he was being treated, came after he was shot in the head - reportedly by a sniper - while talking to journalists inside the perimeter of the protest zone on Thursday.
Al Jazeera's Aela Callan, reporting from Bangkok, said while Seh Daeng's death could inflame the emotions of hardcore protesters, it would probably be a blow to the morale of most of the red shirts.
The government has already rejected a call by the red shirts for UN-mediated talks.
Al Jazeera's Tony Birtley, also reporting from Bangkok on Monday morning, said red shirts he had spoken to were insisting that they would continue their protests despite the deadly clashes with the military.
There was a lot of firing of weapons and some explosions overnight in certain areas of the city, our correspondents on the ground said.
Rising death toll
Besides Seh Daeng, two people were reported killed overnight, bringing the number of dead from street battles since Thursday to 36, according to emergency services and local media. At least 244 have also been wounded in the past four days.
In the early hours of Monday morning, guests at Bangkok's Dusit Thani hotel, which overlooks the red shirts' encampment, were evacuated to the basement after the building came under gunfire and was rattled by an explosion.
Guests were later allowed back to their rooms but the hotel said in the morning that it was closing for fear of more attacks.
Al Jazeera's Aela Callan reports on the deadline for women and children to leave
Fire gutted three commercial buildings in another area and Australia became the latest country to shut its embassy due to the "ongoing violent clashes", after the US and Britain closed theirs.
Jatuporn Prompan, one of the red shirt leaders, urged the country's revered king to intervene in the crisis on Sunday, saying he was the "only hope" for an end to the two-month-old crisis.
In a 1992 uprising, King Bhumibol Adulyadej chastised the military and protest leaders, bringing the violence to an end.
But he has so far not publicly commented directly on the ongoing crisis.
A total of 65 people have died and more than 1,600 have been injured in the two months of protests in Thailand, according to government figures.
Days of prolonged fighting and disruption to normal life have taken their toll especially on Bangkok's 12 million residents.
Most shops, hotels and businesses near the protest area are shut and supermarkets outside the protest zone have seen long lines as people rush to stock up on supplies.
The city's two mass transit train lines remain closed and those wanting to travel in the city often have to take long detours to avoid deadly "no-go" zones.
Red shirt demands
The red shirts said on Sunday that they were ready to enter peace talks with the government immediately, but with the caveat that the UN must mediate.
"We want the UN because we don't trust we will receive justice from organisations in Thailand," Nattawut Saikuar, a protest leader, said.
"What's urgent is to stop the deaths of people. Political demands can wait," he said.
|Parts of Bangkok have become 'no-go' zones amid gunfire and explosions [Reuters]
Sean Boonpracong, a red shirt spokesman, told Al Jazeera that "the situation is deteriorating to a point that if the government is smart, it's best they call off the dogs and push the army back to the base - it's getting to be a civil war".
"I don't know how the Thai government expects to have reconciliation plans while all of us are being taken prisoner".
Sean said, however, that the red shirts were "not closing the door to peace".
But the government rejected the demand for UN-mediated talks, with Panitan Wattanayagorn, the official spokesman, saying "no governments allow any organisations to intervene in their internal affairs".
The crisis appeared to be near a resolution last week when the prime minister, Abhisit Vejjajiva, offered to hold elections - a year early - in November if the demonstrators left their protest area.
But hopes were dashed after red-shirt leaders made more demands and Abhisit withdrew his offer.