Daniel: A carbon tax will increase the price of some goods and services, namely, those whose production and consumption do the greatest harm. That is the whole point. As to whether it raises the CPI in general, that is up to the Fed.
You do raise a good point though, about dependence on fossil fuel for revenue. The irony of a carbon tax is that it is automatically temporary. The more effective it is in phasing out production and use of fossil fuels, the more rapidly revenue from the tax will fall