How do I dispose of my herbarium at the end of its life?

Check with your local regulations, but in most places the best disposal method is to empty the oil onto absorbent material like paper towels and then throw the material into the garbage. The flowers can be composted.
Do not pour the oil down the drain.
If any oil gets on your hands, don’t worry. Silicone is a safe, non-irritating oil. Just rinse it off.