I recently cut 1/8" buna-N rubber sheet for a vacuum gasket.
I purchased a 12"x12" sheet from Zoro. McMaster-Carr and Amazon also sells it.
I could not get it to cut all the way through without charring; however, two passes at 100% made it about half way through. I was then able to use a fresh #11 razor knife blade to finish the cut. The blade followed the laser cut path which resulted in a perfect cut. The heat probably altered the properties of the rubber below the cut, since the blade followed the groove surprisingly well. One word of warning, the Buna-N rubber releases a prodigious amount of flammable material and it looks like a torch flame parallel to the surface at the point of laser contact. The air assist is probably blowing the released carbon black that then ignites. Totally different from masking tape catching fire. The rubber surface is not affected by the flame. Not a material to cut without keeping an eye on it.
Neoprene is Chloroprene Polymer and it’s combustible decomposition is highly corrosive. Santoprene has similar properties but decomposes to standard carbon compounds (CO, CO2, Carbon black, other volatile Organics).