The measure was largely symbolic, however, since parliament is expected to give its approval and most of this year's 40 million dollars in funds have already been disbursed by the government.
But it was seen as a further bid to chip away at the credibility of the tribunal and discredit any of its decisions, at least on a local and regional level.
"As far as Hezbollah is concerned, if the Lebanese government decides that the tribunal is null and void, that would be an important step in the right direction," said Hilal Khashan of the American University of Beirut.
"That way, the tribunal will not have the mechanism for implementing its decisions, especially as far as apprehending indictees in Lebanon," the political science professor added.
The tribunal was created to try suspects in the 2005 assassination of Hariri, whose killing triggered international pressure that forced Syria to end its 29-year military presence in Lebanon.
In a stunning reversal earlier this month, Hariri's son and current premier, Saad Hariri, said he had been wrong to blame Damascus for his father's death.
His about-turn was interpreted as a sign of his Western- and Saudi-backed alliance's weakness in the face of Syria's bid to regain influence over its smaller neighbour.
"Saad Hariri made a colossal concession that was tantamount to an earthquake when he exonerated the Syrians of his father's blood," Khashan said.
"And what you see happening now in Lebanon are the aftershocks which can be as disastrous as the earthquake," he added.
One aftershock came this week in the form of a fierce attack on Hariri by a Lebanese former general jailed for four years without charge in connection with the killing.
Brigadier-General Jamil Sayyed accused Hariri of selling his father's blood in order to frame Syria for the killing, and urged the Lebanese people to topple his government.
Sayyed's comments prompted the country's top prosecutor to summon him for questioning.
Shafik Masri, an expert in international law, said it was clear that the polemics surrounding the tribunal were an attempt to confuse the political and legal issues at hand.
"The tribunal is an international institution in which no one can interfere," Masri said. "But no one knows where this fireball will go and one can hope it will not be at the expense of the poor Lebanese civilians."
There are fears that should the court indict Hezbollah members, this could lead to a Sunni-Shiite conflict similar to the one that brought the country close to another civil war in May 2008.
"Hezbollah and its allies want the prime minister to change his position on the tribunal, and if he doesn't they can put pressure on him," said Paul Salem, head of the Beirut-based Carnegie Middle East Center.
"They can bring down the government or incite civil unrest," Salem added. "Hariri is in a very, very, very difficult position because his options are limited."
Hezbollah, which is backed by Syria and Iran, has two ministers in Lebanon's unity government and is by far the country's most powerful political and military force.
It is the only faction that refused to surrender its weapons following the end of Lebanon's 1975-1990 civil war, arguing that its arsenal was needed to protect Lebanon against potential Israeli aggression.
Khashan said it was clear that Hezbollah and its allies had the upper hand politically and are calling the shots -- at least for now.
"Saad Hariri has no option now but to continue to make concessions," he said. "He allowed himself to be placed in a situation where the wolf is the judge."
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