The poor spend a considerably larger portion of their income on food, heating, and transportation which will be the 3 areas that suffer the most by a generic carbon tax.
The food and agricultural industries represent most of our carbon emission as it is which will increase food prices.
The poor would more likely to live in housing with inefficient heating burning either fossil fuel or using electric heating, and the poor are more likely to have to commute longer distances and are more likely to be dependent on both older vehicles and vehicles such as trucks and van which produce more carbon than newer efficient cars.
This discrepancy will affect both the poor people in the developed world and developing nation that cannot afford the latest and greatest in new technologies, you won't be driving a Prius in central Africa you'll be driving a jeep from the 1980's.
A flat carbon tax is very similar to VAT which is well known to affect lower income families considerably more than higher income ones which is why countries which use a VAT system rather than sales tax often exempt certain products such as food from it or tax it at a lower rate.
When you spend most of your money on consumables or products which will be used to the point of not being able to be resold you are effectively bearing a more burden than those who spend a much smaller portion of their income on consumables and buy products which hold their value well as VAT is designed to capture the value of any potential future resells unlike a direct sales tax which only taxes a single point of sale.
And if we make the carbon tax either deferential or tradeable (e.g. the old carbon caps) it will only result in developing nations being exploited further, if you can buy a nations carbon cap cheaply you'll do so, if the carbon tax in that nation is only pennies on the dollar compared to yours then you'll move your factories there and pollute at will because that would be the economical way to do business.
There isn't a simple solution and while carbon tax is the easiest there's a good reason why allot of economists fear it even more than they fear global warming.