Can I paint plain particle board shelves?
If the shelves truly are unfinished, it will be fine to prime and then paint them. However, it is not common to have shelving with no surface protection at all (like varnish or veneer).

Put a few small drops of water on a back corner in an inconspicuous spot. If the water soaks right in, you are good to prime and paint. If the water beads up and sits there without soaking in... well then the wood is finished and should probably be well sanded (or stripped and sanded) prior to painting.