The move comes after the poll authorities "remained silent" over the Left's request for postponement of the Charilam poll "until normalcy returned".
Polling could not be held in Charilam on February 18 after the death of the Communist Party of India-Marxist candidate. It was scheduled to be held on March 12.
A CPI-M delegation led by party General Secretary Sitaram Yechury on Saturday visited many post-poll violence hit areas in the state.
"The situation in Charilam assembly constituency is horrifically bad due to excessive reign of terror since the BJP (Bharatiya Janata Party) got the majority in the election on Saturday," Left Front convener Bijan Dhar told the media.
Flanked by the leaders of other Left parties, Mr Dhar said: "The leaders of Left parties have apprised the Election Commission in Delhi and Chief Electoral Officer (CEO) of the Charilam situation on March 8.
"Specific instances of all attacks both on party offices, houses of party members and physical assaults on over 20 party leaders and members along with photographs and videos have been submitted before the poll authorities."
Mr Dhar, also the CPI-M central committee member and party's state secretary, said that party's delegation led by politburo member M.A. Baby had met the Chief Election Commissioner Om Prakash Rawat and other two Election Commissioners in Delhi on Thursday and requested them for postponement of election in Charilam till peace and normalcy was restored in the entire state.
Another delegation led by party's Central Committee member Gautam Das met CEO of Tripura Sriram Taranikanti and raised similar demand.
"Both the EC and the CEO had assured that after having ascertained all facts, a decision would be taken about postponement of poll in the violence-hit Charilam. Unfortunately, both the authorities have remained silent," the Left leader stated.
"More than 11 permanent party offices and two district committee offices of CPI-M and RSP (Revolutionary Socialist Party) were burnt down, many trade union offices forcibly occupied by BJP-IPFT workers in Charilam area and CPI-M candidate Palash Debbarma fled from the area to save his life," Gautam Das told the media.
He said around 70 CPI-M leaders' houses were attacked and 20 party leaders physically assaulted by BJP-IPFT workers whereas all 40 election booth offices of CPI-M in Charilam were burnt down.
Tribal based party Indigenous People's Front of Tripura (IPFT) is the electoral ally of BJP.
Mr Das said several thousand Left Front members and workers had been forcibly evicted from Charilam segment.
Tripura's erstwhile royal family's scion Jishnu Debbarma, who was sworn-in as the Deputy Chief Minister of the new government on Friday, is the BJP nominee from Charilam.
"We urged the administration and the BJP leadership to take adequate measures to stop these post-poll violence immediately in the entire state," Mr Dhar added.
Mr Dhar said ever since the poll results were declared on March 3, around 400 Left offices and frontal organisations were attacked; 85 party offices burnt down; 315 offices were forcibly occupied, 1,704 houses belonging to Left members and supporters attacked and around 200 houses set on fire.
He claimed that 290 shops owned by Left members and supporters were burnt down, 10 rubber gardens destroyed, over 600 Left members and supporters assaulted badly, 10 vehicles damaged. This had forced thousands of persons to take refuge elsewhere in the state.