Russian Foreign Minister Sergei Lavrov, speaking after meeting China's top diplomat, said Moscow and Beijing were committed to "the need to strictly adhere to the norms of international law ... and not to allow their violation".
Obama on Monday threatened "enormous consequences" if his Syrian counterpart used chemical or biological arms or even moved them in a menacing way.
The president used some of his strongest language yet to warn Assad not to use chemical or biological weapons - after Syria acknowledged for the first time that it had such weapons and could use them if foreign countries attacked it.
"We have been very clear to the Assad regime, but also to other players on the ground, that a red line for us is (if) we start seeing a whole bunch of chemical weapons moving around or being utilized," he said. "That would change my calculus."
"We cannot have a situation where chemical or biological weapons are falling into the hands of the wrong people," Obama said, perhaps referring to Lebanon's Shiite Hezbollah group, an Iranian-backed ally of Assad, or to Islamist militants.
Turkey's foreign minister has warned it can accommodate no more than 100,000 refugees and that the United Nations may need to create a "safe zone" within Syria to shelter any beyond that number.
Thousands of refugees
A Turkish official told Reuters on Tuesday that about 2,500 people fleeing violence in Syria had entered Turkey in the preceding 24 hours, most of them entering the southeastern Turkish province of Hatay.
Turkish journalist Mahir Zeynalov reported on Twitter that four Syrian colonels and two captains crossed the border early Wednesday.
Turkey is now sheltering close to 70,000 Syrian refugees and is struggling to accommodate the influx, which rose after a bomb attack near the border killed eight, spreading panic.
In Lebanon, street battles between Sunnis and Alawites continued for a second night running, fueled by conflicting loyalties in the conflict across the border. The BBC reported that seven were killed and more than 70 wounded in the country's second-largest city, Tripoli.
Syrian President Bashar Assad, an Alawite, is battling largely Sunni opposition fighters. Lebanese Prime Minister Najib Mikati, himself a Sunni, appealed to both sides to end the "absurd battle" in Tripoli.
In Syria itself, the army deployed tanks on a ring road surrounding Damascus on Wednesday and shelled southern neighborhoods where rebels operate, in the heaviest bombardment on the capital since the army reasserted control last month, residents said.
At least eight people were killed in the shelling, which was accompanied by an aerial bombardment, on the Kfar Souseh, Daraya, Qadam and Nahr Aisheh neighborhoods, they told Reuters.
Regional news channel Al-Jazeera reported that at least 24 people were killed across the country on Tuesday, among them women and children in Aleppo - the city over which the rebel Free Syrian Army (FSA) claims two-thirds control, and where a Japanese journalist was killed on Monday.
"We now control more than 60 per cent of the city of Aleppo, and each day we take control of new districts," said Abdel Jabbar al-Okaidi, a colonel with the FSA. He went on to list some 30 districts which he claimed were under FSA control, including about half of the embattled neighborhood of Salaheddin.
But a security source in Damascus rejected the claims, according to the AFP news agency, calling them "completely false".
Likened to Iraq invasion
Elsewhere, Assad's forces evacuated two security installations at Albu Kamal on the Iraqi border on Tuesday as rebels made gains in the strategically important area after a week of heavy fighting, a Free Syrian Army official said.
Syria's deputy Prime Minister Qadri Jamil said Obama's talk of action against Syria was media fodder.
Speaking after the news conference held by Russia's Lavrov, Jamil said the West
"Direct military intervention in Syria is impossible because whoever thinks about it ... is heading towards a confrontation wider than Syria's borders," he told a news conference in Damascus.
Meanwhile, the Los Angeles Times reported the concerns of Christians, who make up about 10 per cent of Syria's population. It said Christians fleeing the fighting have detected an increasingly radicalized Islamist strain among the rebels that makes them fear for their future.
Reuters contributed to this report.