Someone correct me if I am wrong, But logic mandates that if there is significant IR passing with the green, you will have a Hotter laser.
IR is a longer wavelength and therefore more damaging.
This will have zero effect on "brightness" since the human eye cannot perceive (see) IR.
I still don't understand the fascination with burning things.... Whatever happened to experimenting with optics, mechanics & physics?
Not exactly...
The term "hotter" doesn't really work here.
A beam of light is not "hot." Lasers burn some materials because that light is absorbed, which is what creates "heat."
The IR leakage does add to the total power of the laser, but not significantly. Generally, IR leakage only accounts for 10-20% of the total light output (or up to 30mW) in green lasers from 5-100mW. This does not increase the laser's burning ability by a measurable amount. IR has a different index of refraction than green, so it is not collimated or focused as well as the green output. Also, the IR spill is usually not in a nice uniform Gaussian beam, but instead it is multimode, distorted, and somewhat blocked by the various optical elements and the aperture.
The term "damaging" is also very vague. If you mean IR is better for burning, that is not the case. How well a laser burns depends on several things including the absorption of the specific wavelength by the material, what the material is (metal, plastic, wood, etc.), power density, total output power...
You are right that brightness is not affected.
The short answer is: No, IR spill does not increase the burning ability of a green laser, nor does it make it brighter.