Is it safe to wear a helmet without a plastic covering?
The foam protects the head from impacts by deforming. The plastic cover obviously protects the foam from wear and aging. However it also serves safety aspects in most helmets:

A good helmet has a smooth outer shape, preferably close to a ball, to allow it to slide over tarmac or other hard surfaces in an accident. The hard and smooth plastic outer shell helps this by reducing friction and it prevents snagging.

A reason why we want the helmet to slide rather brake ones horizontal movement is to prevent rotational acceleration of the skull. Due to its inertia the brain resist that acceleration. This in turn exerts a force on the brain (warning, oversimplification the brain is in large parts mechanically decoupled from the skull by a gap filled with fluid). This so called rotational impact is thought to be an important contribution to concussion (cf Kleiven 2013).