As @coolnclever suggested, speeding up the nuclear fusion in the core by adding mass to the star is a good way to begin with your efforts to put out the star. But if you could remove the mass from the star, you won’t need any of your ‘extinguishing material’ to put it out. As you keep on removing the mass from the surface, the density of the core reduces and at a certain point there won’t be any fusion happening at all.
