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Not done one personally, but I used to do lifecycle assessment (LCA) professionally and I’ve read a fair few about food and drink. With coffee I seem to remember it’s extremely water intensive. The little bit that you boil to make your cup of coffee is just a tiny fraction of the total water used in the process.
Paper filters, like any paper product, are quite energy intensive for what they are and very water intensive in production. If they’re bleached, there are also environmental issues with chlorine if an oxy bleach is not used. In terms of waste, they will just compost with food waste at any municipal composting plant so it’s not an issue.
My personal opinion is it’s a small addition to an already intensive flow, which already uses a lot of energy and water. I’d suggest that if a filter paper weighs 2-3g, it probably adds 5-10% to the total environmental impact, tops.
Has anyone ever done an environmental impact breakdown of a cup of coffee? I reckon the paper filter is the least of your worries. Would be interesting to know though.