Prime Minister Han has taken over from the suspended Yoon, who was impeached on December 14 and faces a Constitutional Court review on whether to oust him.
With a majority in parliament, the opposition Democratic Party (DP) passed a bill to appoint a special counsel to pursue charges of insurrection, among others, against the conservative Yoon and to investigate his wife over a luxury bag scandal and other allegations.
The party, which has accused Han of aiding Yoon's martial law attempt and reported him to police, said it would "immediately initiate impeachment proceedings" against the acting president if the legislation was not promulgated by Tuesday.
"The delays show that the prime minister has no intention of complying with the constitution, and it is tantamount to admitting that he is acting as a proxy for the insurgent," Democratic Party floor leader Park Chan-dae told a party meeting on Monday, referring to Yoon.
Han is a technocrat who has held leadership roles in South Korean politics for 30 years under conservative and liberal presidents.
Yoon appointed him prime minister in 2022.
Prime Minister Han Duck-soo has taken over as president from the suspended Yoon Suk-yeol. (AP PHOTO)
Han's office could not immediately be reached for comment.
He has previously said he had tried to block Yoon's martial law declaration, but apologised for failing to do so.
Park also accused Yoon of hampering the Constitutional Court trial by repeatedly refusing to accept court documents, intended primarily to notify Yoon of the beginning of the trial.
"Any delay in the investigation and impeachment trials is an extension of the insurrection and an act of plotting a second one," Park said.
Yoon's ruling People Power Party accused the DP of employing what it called "impeachment politics" and taking Han's political fate "hostage" to push through the special counsel despite multiple ongoing investigations.
A Constitutional Court spokesperson said on Monday it had decided to consider the presidential office's refusal to receive the documents as delivery completed, and proceed with the trial.
A joint investigative team including police and the Corruption Investigation Office for High-ranking Officials has made a second attempt to call Yoon in for questioning on December 25, though it was unclear whether he would appear.
Seok Dong-hyeon, a lawyer helping form Yoon's defence team, told reporters on Monday that he was not delaying the trial but needed time to prepare, and would respond to any investigations after a decision on the impeachment was made.
Woo Jong-soo, investigation chief of the national police agency, told parliament on Monday that police had tried to raid Yoon's office twice but the presidential security service denied them entry.
Woo said his team sent a request to preserve evidence, including a secure phone server.
