Not as much as it used to be, according to my colleague at the New York Times, Austan Goolsbee (The Times owns About.com). See: A Country Less Dependent on Oil Is Free to Make Other New Year’s Resolutions. The relationship between the decline of manufacturing in the United States and the reliance on foreign oil is an interesting one.
This has implications for the ideas put forth by the Pigou Club - a rise in the price of gasoline due to a increase in taxes may be less harmful to the economy than many believe. However, since the nature of oil demand in the United States has changed, older studies which estimate the price elasticity of demand of gasoline may be misleading and raising the tax rate on gasoline may have greater (or lesser) impact on consumption and government revenue than we current anticipate.
More resources on the topic: Gas Tax FAQ, What's the Price Elasticity of Demand for Gasoline?, The Gas Tax - Answering NoPigou's Questions.
This has implications for the ideas put forth by the Pigou Club - a rise in the price of gasoline due to a increase in taxes may be less harmful to the economy than many believe. However, since the nature of oil demand in the United States has changed, older studies which estimate the price elasticity of demand of gasoline may be misleading and raising the tax rate on gasoline may have greater (or lesser) impact on consumption and government revenue than we current anticipate.
More resources on the topic: Gas Tax FAQ, What's the Price Elasticity of Demand for Gasoline?, The Gas Tax - Answering NoPigou's Questions.

Comments
I think the u.s should us oil in the u.s.