If the supply curve is perfectly inelastic, then the tax incidence on consumers is zero; and the tax will have no impact on equilibrium price or quantity!
This is an unusual situation, but a supply curve that is perfectly inelastic with respect to price is vertical. This means that producers are willing to supply the same quantity at any price (at least within the range of concern).
Here is a graph to help explain:
In the graph, P(1) is the initial equilibrium price. Because producers will supply the same quantity at every price, the only impact of this tax is to reduce producer surplus. After the tax, producers receive P(2) -- which is just P(1) - $1. Consumers still pay P(1), Equilibrium quantity does not change. In other words, producers effectivly pay the entire tax, regardless of how it is levied.
Consumer surplus does not change. Producer surplus is reduced by the amount of the rectangle bounded by P(1) and P(2), the vertical axis and the equilibrium quantity (not labeled - sorry! - I just found FlockDraw to create this image freehand).
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