Is it safe to wear a helmet without a plastic covering? A friend of mine gave me a foam bike helmet had the plastic covering removed from it. Besides for the plastic being removed, the helmet is in excellent condition. Is there any issues with using this helmet for riding locally as far as safety is concerned? I think the plastic cover was thin to begin with so I think it wouldn't make much of a difference.
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).