Quran Burning is a specific type of book burning. It is an act of extreme aggression against Islam and is also one of the most hateful acts a person can commit towards a Muslim. The Quran is considered the most sacred text in the Islam world and burning it is a sacrilege so great it is often punished by death in countries that are ruled by traditional Muslim Law.
Even in countries where it is not punished by death the burning of the Quran remains an immensely controversial act that is seen as hatemongering by many - it also incites Islamic Fundamentalists into revenge attacks and is as disrespectful to the Muslim world as Bible Burning is to the Christian world.
Due to its immense stigma, the act of burning the Quran (at least publicly) is extremely rare - the most infamous example was the Westboro Baptist Church, who burned a Quran in front of a Muslim and taunted them over it following September 11 attacks. Another person who has done this is Fundamentalist Pastor Terry Jones.