Remember that the Tyranids you see on the battlefield never evolved in the conventional sense, but were specifically engineered for combat by the Norn Queens. They only possess exactly enough digestive system to make them as lethal as possible, leaving the actual planetary-scale digestion to later parts of the swarm. As with most 'nid things, it depends on the individual bioform.

The smallest tyranids (eg Rippers) don't poop, since their job is to gnaw the biosphere down to the bedrock and return to reclamation pools anytime they're full. They're more ambulatory stomachs than true creatures really.

Termagaunts and Hormagaunts have very simplistic digestive systems, designed to quickly extract enough energy from what they consume to continue attacking whatever is in their path. They don't need to extract complicated nutrients, because they're not designed to last very long. If they need particular proteins to self-repair, they can simply eat the bodies of their fallen comrades. They'd probably poop a horrid half-digested slop studded with boneshards, bark, shreds of flak-armour, and anything else that doesn't break down quickly. Or perhaps cough up pellets like an owl. As with many animals, they'll probably eat their own poop to send it through the digestive tract a second time if needed.

Larger tyranids such as Warriors are more valuable, are expected to have a much longer lifespan, so would likely have a commensurably larger digestive system. They may well get injured in the field and need to repair heavy wounds, so being able to extract and store all kinds of nutrients would be very valuable. It's possible that smaller tyranids might act as nurse-maids to any injured warriors, bringing them fresh kills, vomiting up half-digested meals, or even allowing the warrior to consume them. Warriors likely poop something similar to most animals, as they will extract everything useful from their meal, without bothering to spend the time and energy to break down the toughest substances such as bone or steel.

Siege-breaker Tyranids such as Carnifexes are designed to smash and consume the largest and toughest enemies. This requires a vast quantity of energy to perform constantly, so they must have very dense internal energy stores. It may even be that they would require an unsustainable volume of sugar-dense foods to meet this need (like a humming bird), and in fact they must get their energy from more esoteric means; the fact that many spit bio-plasma supports this theory. In that case, any standard digestive system would likely be a holdover from when they were being grown, or would be tuned to breaking down the hard materials (eg steel, ceramite) that the 'fex swallows. I doubt they would poop much, if at all.

Zoanthropes are sustained solely by pyschic energies, and their digestive system (along with the rest of their bodies) is entirely vestigial.

Flying Tyranids such as Harpies are able to quickly return to tyranid-infested areas, or even Hive Ships in low-orbit. I doubt they have much of a digestive system (as that would increase weight needlessly), instead being sustained by regularly returning to suckle at some kind of hive or roost. If they need to feed in the field they likely have much the same process as Gaunts, extracting the energy they need quickly and letting the remainder be "released to the wind".

Infiltration organisms such as Genestealers will have quite complicated digestive tracts, since they're designed to operate alone for decades at a time. They'd likely poop like normal animals. Lictors would have the most complicated digestive systems of all, since they're designed to extract not just nutrients but information and DNA from their victims. And as their mission is incredibly stealthy, they likely poop hardly at all and would take great pains to hide any necessary emissions.

Bio-titans are at a completely different scale. I have no idea what it would take to sustain such a creature, but it probably doesn't involve anything like eating and pooping.