Will laser paper cutting burn?

no, the focal point of the laser is very small. In order for the paper to burn completely, it must be that the heat released by the ignited paper can heat the surrounding environment to the ignition point of the paper, so that other parts of the paper can burn. The laser focal point is too small, and the size of the paper lit at one time is limited. The heat released by the burning of those papers is not enough to heat the surrounding paper to reach the ignition point, so it will not burn.