I've been told...heat a pin in a flame and press the tip into the material. Ivory will not melt/yield, plastic will. There may be some synthetics that don't melt either though.
Genuine ivory and bone both appear bright white under a black light, whereas synthetic materials tend to absorb more light and appear dull.
Look for pores to confirm the item is bone.
Examine for lines to determine that your piece is ivory. When you examine your piece under a magnifying glass, if you are looking at ivory, there will be lines on the surface. The lines on genuine ivory can appear in a variety of ways, including straight, cross-hatched, or in a circular motion.