I’ve seen estimates of a billion cards going in the bin after Christmas. I’ve seen estimates of several hundred thousand trees needing to be felled to make those cards in the first place. I’ve not tracked down any definite figures online, but it doesn’t take much thinking to consider how many cards might be sent, and what it took to create them and come to the conclusion that it’s a very high environmental cost, regardless of the precise figures.
At this point, you may have cards. How you dispose of them will make a difference.
You could cut them down for reuse, as gift labels. You could use them in future crafting projects. You could give them to a charity that can make money recycling them. You could recycle them if they aren’t covered in non-recyclable things.
For the longer term, you can think about buying recycled cards to reduce impact on trees. Buy smaller cards that use less material and create less waste. Consider not giving cards.
Cards are an example of things we think of us just a bit of seasonal fun and goodwill. How much harm is it ok to cause for the sake of a bit of fun and festivity?