I spread the coffee grounds around outside camp in the bushes. They're about as natural as you can get after the coffee has been boiled out. Just pack out the filter paper, and if you shake the grounds off them and rinse they can be reused several times to brew more and they burn well when dry.
I had a bear one time grab my zip lock full of ground coffee, but he spat it out. Hard to imagine anything wanting to eat coffee grounds beside maybe an earthworm or a bacteria.
However with all fairness, I do not exactly believe in the value of the no trace ethic. I think many portions of the LNT concept are highly flawed and extremely "area of use" defined. In other words, on the appalation trail in crowded eastern states you would follow different guidelines than on the Pct say through Oregon. A nice premade campsite in Oregon would prevent a LOT of people from trying to scratch out a tent site, while covering up camp sites without tools is very difficult. Having to make a camp site then camoflaging it, every day, would be an unreasonable burden on a through hiker. I believe some "improvements" are improvements not detriments to the wilderness. A rock pile wind break on a summit will be used as shelter by perhaps hundreds of people.
These are my own opinions based on wisdom earned through many wrong decisions. Your mileage may vary.