The answer is a definitive - maybe. Or - occasionally. Or - as needed.
While the question seems simple, an answer depends on too many variables.
On the whole, carbon isn't needed in the filter of a healthy planted tank.
As to carbon striping fertilizer from the water, I've challenged people to provide something other than anecdotal tales, and gotten no citations of any research supporting the theory. I work in a university research library. I have access to subject librarians. The university happens to have a fish lab and we happen to have more fish related literature and journal access than most.
There is a lot of data on what carbon or charcoal will do to or for water in tanks. There is lots of data about what happens to carbon over time, in tank filters. The important take away from those studies is that carbon stops being useful after some period of time. A time period that varies depending on the type of carbon and what it is derived from.
If you want a personal take, I use an AmmoCarb mix in new tank setups if I'm going to put fish in before I'm sure the tank has fully cycled. For example, I have a fresh tank setup with filter squeezings from a mature filter and a used sponge in the AquaClear filter. I'll be moving mature air driven sponge filters in there as well.
Tomorrow, I'll be putting a couple of dozen CPDs in that tank. A bag of AmmoCarb in the filter basket and a few drops of Prime each day will help ensure the health of the fish. There will also be a piece of Poly Bio Marine Poly-Filter® in there so I can visually monitor the water quality.
After a couple of weeks, I'll pull the bag of AmmoCarb.