If the demand for a good is elastic it means a change in price would affect a change in quantity demanded.
Now if the demand for that good is more elastic than its supply, the tax burden is borne by the producer. This is because, when there is an increase in price it will lead to a decrease in the quantity demanded, consumers would most likely move on to consume a substitute, hence the producer of that product would be left to bear the burden of any other additional increment in taxes or running cost.