Name the play during the performance of which the Globe Theatre was burned down in 1613.

The correct option is Henry VIII.

The Globe Theatre, originally built in 1599 by William Shakespeare's playing company (the Lord Chamberlain's Men), met a dramatic end on June 29, 1613. During a performance of the historical play Henry VIII (also known by its alternative title, All Is True), a theatrical special effect went wrong.
To create a realistic and grand atmosphere for the play, a small cannon (a chamber) was fired during Act 1, Scene 4 to mark the arrival of King Henry VIII at Cardinal Wolsey's house. However, the sparks and hot wadding from the cannon shot landed directly onto the thatched roof of the circular wooden structure.

The dry thatch caught fire immediately, and the flames quickly swept around the entire building. Despite the theatre being packed with spectators, the building was evacuated successfully. In less than two hours, the original Globe Theatre was completely destroyed by the fire. The only reported casualty was a man whose trousers caught fire, which was luckily put out with a bottle of ale.
Following this disaster, the Globe Theatre was rebuilt on the same site and reopened in 1614, this time featuring a safer tiled roof rather than a thatched one.