They accuse Mr Samak of being a puppet of former Prime Minister Thaksin Shinawatra, who was ousted by a military coup in 2006 and is now in exile despite facing trial over corruption charges.
The protests began on Tuesday, when more than 30,000 people took to the streets in a well-coordinated action.
The demonstrations have been organised by a group calling itself the People's Alliance for Democracy (PAD) - formed three years ago as a movement to bring down Mr Thaksin.
Mr Samak has repeatedly changed his approach to the protesters - prevaricating between advocating a "softly, softly" approach and vowing to take a harder line.
In his latest comments, he said he had the weight of the law on his side, but wanted to avoid bloodshed.
We won't leave Government House as ordered by the Civil Court
On Wednesday night, riot police were ordered to clear the protesters out of Government House, after Mr Samak obtained warrants for the arrest of nine top leaders of the protests on charges of insurrection.
But with thousands still occupying the grounds and buildings, and some well-constructed barricades blocking police vehicles from entering the area, the security forces apparently decided it would be impossible to carry out their orders without bloodshed.
Shortly after sunrise, the riot police suddenly pulled back.
Makeshift weapons
The Bangkok Civil Court has now issued a ruling ordering the protesters to vacate the government compound.
But about 10,000 protesters are said to remain at the protest site, and the PAD leadership say they plan to appeal the ruling.
Occupying protesters have erected makeshift barricades
Many protesters are armed with makeshift weapons including golf clubs and bamboo sticks, and some have formed a human chain around the group's top leaders as well as barricades around the site.
"We won't leave Government House as ordered by the Civil Court," one PAD leader, Chamlong Srimuang, told reporters according to Reuters news agency.
"Our demands remain the same - to have the government resign and to prevent an amendment of the 2007 constitution," he added.
The PAD fears constitutional amendments might offer a legal justification for throwing out the charges against Mr Thaksin.
Mr Chamlong - an ascetic Buddhist and retired military general - urged supporters not to abandon the site if he were arrested.
Conditions inside the occupied compound are said to be deteriorating, with rubbish piling up, men relieving themselves in the open, and laundry hanging from window ledges.
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Reluctant
Mr Samak has vowed not to bow to the PAD demands for his resignation. He points out he was legitimately elected last December, and correspondents say he remains popular with Thailand's rural majority.
But for all his tough talk, the security forces seem reluctant to confront the throng of protesters, says the BBC's Jonathan Head in Bangkok.
Public sympathy for the protesters is wearing thin. However, the PAD has powerful and wealthy backers, and possibly some support within the armed forces, he says.
The army commander says troops will not get involved, but the police may not have sufficient forces and equipment to move such a crowd on their own.
And the longer they remain in control of the prime minister's office, the weaker his position appears to be, our correspondent adds.
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We won't leave Government House as ordered by the Civil Court
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