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This is certainly a legitimate concern, especially when you consider that we've heard for decades that we desperately want to save the forests because trees remove carbon dioxide from the air during photosynthesis, and this can help offset the carbon dioxide in the air naturally. You don't need to worry about that in relation to firewood, because the wood that ends up in your fireplace is not coming from a functioning forest that was leveled just so you could have a warm fire. Firewood in the United States generally comes from the excess of the lumber industry's cutting business or from trees that are being purposely cleared from forests to help either protect against forest fires or to let younger trees thrive from the open space. Most of these trees are then replaced, so the net impact on forest levels is almost zero from firewood-related purposes.
A second issue relating to firewood's environmental impact is the burning itself. I'm sure everyone who has seen a smoky fire thought they must have been damaging the environment in a similar manner to a factory spewing gases, or a car's exhaust pipe. That is just simply not true in this case. The smoke from a fire in your fireplace is natural and doesn't do any appreciable damage because there just simply isn't enough of it to be an issue. This is especially true if you are using dry "seasoned" firewood, which burns with a minimum of smoke. If you're going to burn wet wood every night then you are going to produce a large amount of smoke and potential damage, but dry firewood puts off little smoke at all.
So while your concern is admirable, it's largely unfounded. Grab some firewood out of your log rack and enjoy that nice warm fire!