If you are on city water (sanitized for your protection with many wonderful chemicals) then I would strongly urge you to NOT toss in the compost heap. Further if you have a water softener the salts will kill plants - again, no tossing on the heap. If you are on well water, without a water softener then I would check the labeling on the box and see what is inside that filter before tossing in the heap. charcoals is ok for the compost heap - it has to be broken down and mixed in rather well. Some filters use salts and other minerals which can affect over all plant health and soil health - some which are nontoxic for plants can be toxic for your helpers (like worms) leading to dead soil. Also many areas have very, very low amounts of heavy metals in their water supply (Lead, copper, uranium - salts there of, etc) Although while dilute in water and in your water we are talking parts per billions - something that will not harm the human body in the water - they are concentrated and trapped by filters and over the years you can end up amending your soil with a lot more lead than is healthy for those who eat the plants. Ending up with heavier meals... heavier in lead. I think you already know that lead is not a nice thing in higher quantities. If you are on municipal water a call or a web search will tell you how much of what is in your tap water. You will find that the western world holds very high standards for drinking water. If you are on well water you need to have that checked on a regular basis - if new building is taking place around you then a sample every six months. If not then at least every three years. If the amount of any heavy metal is near the EPA standards but low enough for drinking water, your filters will load up on them faster, each trip to the compost heap will add more heavy metals. In any case if you are using a filter there is something you do not want to drink. Safe to say that whatever that something is you do not want to feed it to your vegetables which will pick it up and it ends up back in your kitchen - granted you are no longer drinking it, your just eating it.

I would not compost with the filter media. You are removing "things" from the water that you do not want to consume with your water and concentrating them on the filter media. That is the type of waste that should go to a landfill and be contained within the highly engineered barriers of the landfill.