The group, which is led by a former Tiger commander, Colonel Karuna, said on Monday it would also hunt down three senior rebels and hand them to Nordic truce monitors.
A Karuna aide said: "Our military wing will kill anyone illegally occupying the homes and businesses of Muslims, whom the Tigers forced to flee Jaffna, if they don't vacate within a month."
He spoke on condition of anonymity after a meeting between leaders of his Tamileela Makkal Viduthalai Puligal (TMVP) movement.
"We will hunt the Tigers responsible for the Muslim exodus. We want them to face charges of crimes against humanity at the Hague," he said.
The Liberation Tigers of Tamil Eelam (LTTE) gave tens of thousands of Muslims only two hours to abandon their properties in the northern Jaffna peninsular in 1990 in what was seen as ethnic cleansing. Only a few hundred have dared to return.
The Tigers had no immediate comment on Karuna's threat.
Karuna ran LTTE military operations in east Sri Lanka until he split in 2004.
"We will hunt the Tigers responsible for the Muslim exodus. We want them to face charges of crimes against humanity at the Hague"
He is now at the centre of a deadlock in efforts to turn a truce into lasting peace after a two-decade war that killed more than 64,000 people.
The Tigers accuse the military of helping Karuna's renegades to attack their fighters. They say they may refuse to continue emergency talks with the government to shore up the ceasefire unless the military disarms their former comrades.
Karuna says his men will disarm only if the Tigers do too, and vows to retaliate if attacked.