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).
One thing the plastic is designed to do (and for which it doesn't need to be thick) is to have low friction with the road. So if you come off your head slides rather than snagging. The friction that the plastic stops would injure your neck, and also lead to very fast abrasion of the foam,perhaps just in time for it to be destroyed before you hit something solid. Even a fairly thin layer backed by foam also spreads the impact of a point over a larger area. Flexible plastic does a very good job of holding the helmet together, so that it absorbs more energy if it splits and still has some use after the first hit (one crash can easily cause multiple head impacts)
You are correct that the impact protection comes from the foam. The plastic covering is there to protect the foam from dents and gouges. Now that it is gone (which I doubt could be done without damage, most helmets are directly molded into the plastic), it will be easily damaged.
Fun fact: Early non-racing helmets didn't have the plastic cover. They generally didn't last long.