If supply is perfectly elastic, then a supplier will only produce the product at a certain price. If the price were to fall, the supply would essentially be zero, as the supplier would not make any money. This means that the producer is driven to pass the tax burden off to the buyer in order to maintain their profit. If the producer keeps the tax burden, they will go out of business because they have no "wiggle room" with the price of their product.
Assuming that government regulations allow the producer to do so, they will therefore pass off their own tax burdens in order to maintain their ability to stay in business as a producer in the perfectly elastic supply scenario.