Will plaster of Paris dissolve if used inside a home made snow globe?

Yes, so the plaster will need some type of water-proof coating.

If you are carving something from a plaster block, moulding something out of plaster, or even using a pre-made plaster object, the only way it will not slowly dissolve in the water of the snow globe, is to make sure it is totally covered in some type of water-proof finish. I'd use a minimum of 2 coats to make sure. If there is even a small break in the finish for the distilled water to get under, your finish will begin to peel and the plaster will break down there.