I'm not sure about the melting of rubber/neoprene components of a lawnmower engine since they are usually of the same material as a car.
But, premium gas will make the air-cooled lawnmower engine run hotter, in most cases beyond the normal operating temperature. Since most of these engines rely on lubrication by splashing the oil around rather than an oil pump, the hotter oil will loose its viscosity, and the engine might fail. These engines run hotter than water-cooled counterparts normally, so the added heat may be enough to melt the rubber/plastic components where they come in direct contact with the metal motor.
While we're on the subject, this is also the reason most of these engine manufacturers specify straight 30w or 40w oil instead of the multi-blends found in cars. The multi-weight oil tends to not work well in splash-lube designs.