Incorporating biological waste into Dwarf Fortress should do so in a way that meshes with the pre-existing texture of the game. It should provide an increase in entertaining complexity. It should not gross out players who don't want to wallow in it. It should take advantage of pre-existing mechanics wherever possible. There should be multiple paths to dealing with it that do not feel contrived. I've seen a lot of the following suggestions in this and other threads . . . it's a compilation of much deep thought by many individuals.
So, first off-words. Language should be reasonably frank but emotionally neutral . . . wherever possible, it should avoid the extremes of rudeness (at one end) and clinical terminology (at the other). For solid waste I would advocate "filth" or even "dung" over "poop" or "shit". Liquid waste is probably best left as "urine".
Manifestation in game. Filth should be an item that decomposes over time. Dwarves should get an unhappy thought from having to handle it or step in it. Urine should be a liquid subject to evaporation, and dwarves should also dislike stepping in it. Each of these should produce miasma indoors after a bit (maybe smaller amounts of it for urine) and possibly attract pests such as flies. There's no particular reason to have dwarves track either substance around. The presence of either substance in a body of water causes nearby water to be murky.
Production. Dwarves should produce units of filth per year in step with the units of food they eat, and should produce urine per year in step with the units of booze or water. Time between eating/drinking and the need to excrete could be variable or based on a statistic like constitution. A dwarf who's gotta go but can't for some reason should get a bad thought. When animals are instilled with the need to eat and drink, they'll operate under the same circumstances . . . until then, they produce droppings randomly.
Natural disposal. Urine will evaporate from a hard surface after a while, and will quickly soak away into a porous surface (such as soil). Filth will turn into soil (suitable for plant growth). While it is transitioning, if it is in water, it might turn areas of water murky.
Deliberate disposal. The player may identify a zone as a waste dump, which is used directly or into which buckets of filth are dumped. A dwarf might get a mildly unhappy thought about the state of 'the facilities' for using a bucket or the unmodified dump zone. More elaborate zones might consist of a grate over a 1x1 pit. A single new building ('commode') might even provide a mild happy thought depending on the quality of its construction (Urist was pleased with the quality of the facilities lately) and sans pit might be equipped with a bucket for later dumping. Healthcare workers might provide buckets to bedridden, chained, and otherwise immobile dwarves when it is required (unattended dwarves might unhappily lie in their own filth).
Dwarves with the 'cleaning' labor enabled take buckets of filth and urine to the dump zone. They also collect filth and urine deposited on the floor by animals and the insane (the latter won't necessarily be intrusive as insane dwarves seem to be killed quickly or don't eat/drink).
Waste-based industry. Waste can be converted into fertilizer at an alchemist's shop for use on fields and by planters to (b)uild arable surfaces on bare rock, if the dump zone is accessible. Filth and urine become substrates required by tanneries, unobtrusively sought by tanners.
Advantages to this system. The game becomes more complicated, but in a non-overwhelming way. There is a reason to build an elaborate sewer system, but it is not necessary. Players who don't want to deal with it much at all can designate a hole in the ground and not have to interact with poop much beyond that. Those who want to can add a gold commode to every noble abode. Waste provides a source of fertilizer for fields (not necessarily necessary, but nice to have) and makes tanneries appropriately miasmic. There is a reason to segregate animals to specific areas of the fortress, and if you do happen to have cows and dogs and kittens running in uncontrolled mayhem all over, the fortress will become filthy (as it should . . . pet peeve of mine).
Players who don't want to deal with it are not onerously exposed to scatology, while those who really can't see the world any other way can dunk goblins and elves into cesspits to their hearts' content.