One way is with a hot pin, this will tell you if it is plastic. A pin hot enough to burn pushed on a discrete place of an item in question when applied will soon tell you if it's plastic from the smell of the smoke.
This will not differentiate it from bone. This can be done as ivory has a distinct very fine grain (as with woods). Bone however does not and often has small holes left where blood vessels were, whereas ivory has no blood vessels through it. It can still be difficult where niether of these distinctions are apparent and is difficult to demonstrate without pictorial examples to show.