Simple. Some people are allergic to certain metals, mainly nickel, contained in the mouthpiece. Whilst we're on the subject, we have a horn player who is so afflicted. In the abscence of a horn mouthpiece from Kelly we were discussing at band the other night the possibility of lathing a new mouthpiece from a solid block of acrylic. Anyone ever attempted such a feat of engineering before?
Many marching band brass players use plastic mouthpieces (either just rims or entire mouthpieces) when they must play in cold weather.
It's possible that a proper mouthpiece for concert work could be made from non-metals, however, it would take a bit of engineering. Part of the sound of a brass instrument is created by the mass and balance of the material in the mouthpiece. For example, one problem with the marching band plastic mouthpieces is that the plastic changes with temperature in a different manner than the metal of the instrument, which can lead to air leaks or a stuck mouthpiece.
A better choice than regular acrylic might be an advanced ceramic, which would have the advantage of being very smooth and also non-allergenic. I could also see advanced ceramic applications in the valves (maybe replacing the plastic guides found in many modern valves, which have to be replaced often because of their wear characteristics).