I knew sooner or later someone would write this paper

Yes Pigou taxes can possibly have counterintuitive results:

Judged by the principle of intertemporal Pareto optimality, insecure property rights and the greenhouse effect both imply overly rapid extraction of fossil carbon resources.  A gradual expansion of demand-reducing public policies -- such as increasing ad-valorem taxes on carbon consumption or increasing subsidies for replacement technologies -- may exacerbate the problem as it gives resource owners the incentive to avoid future price reductions by anticipating their sales.  Useful policies instead involve sequestration, afforestation, stabilization of property rights and emissions trading.  Among the public finance measures, constant unit carbon taxes and source taxes on capital income for resource owners stand out.

Here is the full paper, here are non-gated versions.

The point is this: carbon taxes drive down the net post-tax price which fossil fuel suppliers receive; in fact the resulting transfer of wealth from the Saudis to the U.S. is a common argument for such taxes.  But the lower net supply price is not in every way a blessing.

One possibility is that at the new and lower (supply) price, other (non-Pigou-taxing) countries will buy more fossil fuels, picking up some of the slack and counteracting the carbon taxes at the global level.  This effect is strongest for low marginal cost oil.  If Americans buy less oil but all the oil will end up sold in any case, demand simply has been redistributed rather than lowered.  Instead the key is to get that oil to stay in the ground.

Second, at a lower supply price the intertemporal Hotelling resource extraction problem is tricky rather than straightforward, again especially for low marginal cost oil.  Lower (supply) prices today also mean lower (supply) prices in the future, so, after a tax is imposed, the time path of extraction can easily tilt toward the present rather than away from it.

Note that the higher the (posted) oil price goes, the lower are extraction costs relative to price and the more likely these counterintuitive results become a potential problem.  In 2006 average extraction costs were only 15% of the average spot price.  Obviously if extraction costs are high the lower net supply price to the producer does keep the stuff in the ground.

Further counterintuitive results can arise if market players expect a Pigouvian tax to rise over time.  A (politically impractical) alternative is to make the tax very high today and lower it over time.  We are more likely to do the opposite.

The bottom line is this: paying countries to blow up their oil fields may be more effective than taxing the resource.  (TC: don't some people volunteer to do this work for free?)

Empirically, I still suspect that the "naive" first-order demand side effects predominate.  But if you want to know all the ins and outs of Pigou taxes, it's worth spending some time thinking through these problems.  Taxing intertemporal resource stocks doesn't always lead to simple, predictable results.