At least four deaths and dozens of injuries were reported and homes and shops were torched, police said.
Mollah, who had been assistant secretary general of the now-banned Jamaat-e-Islami party, was hanged Thursday night after being convicted of war crimes during Bangladesh's 1971 war of independence from Pakistan.
The execution added another dimension to election-related violence involving various opposition parties led by the main Bangladesh Nationalist Party and the government of Prime Minister Sheik Hasina.
Mollah was buried in his village home in Faridpur in central Bangladesh early Friday. The funeral was attended by about 500 people of the village, the report said.
The overall death toll in weeks of violence was about 40, officials said.
The Bangladesh Daily Star reported at least four people died in clashes between supporters of Mollah and activists of the ruling Awami League party, who welcomed the hanging. The deaths were reported in Satkhira and Pirojpur towns.
The report, quoting a police official, said activists of Jamaat-e-Islami and its student wing vandalized, looted and burned homes and businesses of Awami League supporters.
A report by bdnews24.com said there were several clashes in other parts of the country, including in Monoharganj where police said Jamaat activists tried to pull out railroad tracks.
In Upazila town, police said at least 20 shops were vandalized and about 20 people, including three policemen, were injured.
Dozens more were injured in clashes in other towns and cities, the report said.
Opposition parties have rejected the government's decision to hold elections Jan. 5. Instead, the parties want a neutral caretaker government to conduct the elections.
The Awami League came to power in 2009 after a landslide victory.