Using alder cones

If the tannins (stain) from Alder cones is what you are interested in getting by adding to aquarium, then the visible aspect of these tannins will likely be taken out with carbon. But, the acidifying/antibacterial properties of alder cones will still be predominately active in tank water. In other words, while some of the benefits will be taken out through carbon filtering, a good amount will not be.
But, Alder cones have many other valuable benefits beyond the release of tannins: they also provide surface area for beneficial bacteria/ biofilm and act as a surface for the development of supplementary food source for freshwater shrimp, algae eaters like otocinclus, and fish fry.

I use carbon when I first set up a tank during cycling, when I am taking out medications from water column, or have issues with suspended materials in water. But, the majority of the time a cycled, well filtered, maintained aquarium can do without it.

 

That is sensible. Throw the charcoal out and if you think you will need it in future buy a new bag. Charcoal gets exhausted after about a month and needs to be replaced to be effective.