Engulf or Envelop
You can use "engulf" or "envelop" to describe the action of going from a regular state to a state where the monster has fire on it.
engulf: to surround and cover something or someone completely
envelop: to cover or surround something completely
There is an ability called "Flame Body" in Pokémon (think Magmar, Rapidash, etc.), where any contact burns the other Pokémon. Other Pokémon that have fire on them include Charizard and Infernape.
In Fantastic Four, Johnny, aka the Human Torch, has this ability and the power to manipulate fire.
[Johnny] gained his powers on a spacecraft bombarded by cosmic rays. He can engulf his entire body in flames, fly, absorb fire harmlessly into his own body, and control any nearby fire by sheer force of will. "Flame on!," which the Torch customarily shouts when activating his full-body flame effect, has become his catchphrase. - Wikipedia (Human Torch)
The Human Torch possesses the physical ability to envelop his entire body or portions of his body with fiery plasma without harm to himself ... - Marvel Database
So you can say the following:
When the monster gets upset, it envelops its entire body with flames.
When the monster gets upset, it engulfs its entire body in flames.
If the flame is on a very specific part of its body, you can use "burns":
The tip of its long, tapering tail burns with a sizable flame. - Bulbagarden on Charizard