Tungipara municipal Chhatra Dal and Jubo Dal erected a memorial arch to mark the second anniversary of the July movement.
The arch was built on the afternoon of 1 August near Gate No. 1 of the Bangabandhu Sheikh Mujibur Rahman mausoleum complex in Tungipara, Gopalganj.
Organisers said it commemorated people killed and injured during the July movement and was part of anniversary programmes.
Witnesses said unidentified people set the structure on fire later that night before leaving the area.
They said the lower part of the arch was partially burned.
Tungipara police station officer-in-charge Ayub Ali said officers visited the site after being informed of the incident.
Ayub Ali said police were investigating who was responsible.
He said no written complaint had been lodged with the police station as of Sunday afternoon, and any further action would follow directions from higher authorities.