Power is power. 100mW of blue/violet light is EXACTLY the same power as 100mW of red light. How that light is absorbed into a material depends on the material, and that will affect burning, but power is power, 100mW is 100mW.
What you read is true, a single photon of blue/violet has more energy than a single photon of red. So if a 100mW violet beam is the same power as a 100mW red beam, then it is clear there must be fewer violet photons "in" the violet beam than red photons "in" the red beam. Each violet photon has more energy than a red photon, so you need fewer violet photons to have the same power in a laser beam. But the power of a laser is the total of all photons, so 100mW is 100mW, no matter what the color is, so the burning ability is exactly equal for both colors.
If your target absorbs colors differently, then burning ability will be different, but not directly because of what you're talking about (absorption is related to photon energy, but in a more convoluted manner than is worth discussing here).