The most powerful argument against a carbon tax is that it raises the prices of just about everything in an economy. This is particularly a bad idea if one country imposes a carbon tax on itself while other countries do not.
A carbon tax increases the cost of essentially everything in an economy. This is because everything is produced using energy. A carbon tax would increase the price of the energy used to make a product in a factory. It would increase the price of the fuel used to transport that product to the market. It would increase the price of the electricity used on the lights and the air conditioning in the market.
Imagine, then, if one country imposed such a tax while others did not. All of the products produced by that country would increase in price. In the mean time, those of other countries would not increase because there would be no carbon tax in those countries. This would put the first country at a terrible disadvantage in terms of trade.
Thus, the main problem with a carbon tax is the fact that it raises the cost of living by raising the price of everything that is produced through the use of energy.