The BBC understands this incident happened in April. Palazzo Maffei released the footage on 12 June.
The chair was built by Italian artist Nicola Bolla and is bejewelled with Swarovski crystals made from polished, machine-cut glass. It is named after Vincent van Gogh as a tribute to the Dutch artist's .
Bolla's piece is somewhat priceless, in that the museum declined to provide an estimate of its value when asked by the BBC.
Carlotta Menegazzo, an art historian based at the Palazzo Maffei, says that - while it looks sturdy - its frame is mostly hollow and kept together with foil.
"On the chair was a note warning people not to touch, and of course it is placed on a pedestal, so it's quite clear it's not a real chair," says Ms Menegazzo.
Two legs and the main seat were broken, but Ms Menegazzo says "a great job" has been done to restore the piece and it is now back in place.
The Palazzo Maffei opened in 2020 and has 650 pieces on display, including paintings by Picasso and ancient Egyptian art.
Ms Carlon says the majority of visitors are considerate, and she hopes this release of CCTV footage won't become a "negative episode".
Instead, she wants to highlight that "anyone should enter art places, or museums or churches, wherever art is displayed, in a more respectful way".
"Art must be respected and loved because it is very fragile," she adds.