What sort of paint do you use to paint over plastic?

My best advice is to remove the plastic. However, if the plastic is securely fixed to a wall you wish to paint, you may find a decent result by first sanding the plastic. The point is made that the surface must be roughed up a bit to better ensure paint adhesion.

If you are satisfied you can sufficiently rough up all of the plastic you plan to paint over then use a primer coat of latex paint over which you will apply additional coats of latex finishing paint. DO NOT attempt to use an oil base paint as the adhesion will be poor and the oil within the paint will damage the plastic resulting in peeling.