Are particles in curved spacetime still classified by irreducible representations of the Poincare group?

What is a particle?

Before we worry about how to classify particles, we should try to be clear about what "particle" means. Ideally, we would like the definition to have these features:

• Particles can be localized.

• Particles can be counted.

• The vacuum state has none of them.

Even in flat spacetime, this wish-list already has a problem. In relativistic QFT, the vacuum state is entangled with respect to location, as is any other physically reasonable state. Observables are localized, but states are not. The idea that a state can be "identical to the vacuum state outside a given region of space" is only approximately meaningful. A physically reasonable deviation from the vacuum state, such as a particle, cannot be strictly localized in any bounded region of space.

In flat spacetime, we can work around this by conceding that particles can only be approximately localized. Then the requirements about counting can still be exactly enforced.

If we have a large enough gap between the curvature scale and the approximate localization scale, then we can still use this in curved spacetime, but in that case we have another problem: what do we mean by "vacuum state"? If the spacetime is stationary, then we have a preferred time coordinate(s) and can define the vacuum state as the state of lowest energy, as usual. But even in a stationary spacetime, different local observers may perceive different states as being locally vacuum-like, with due respect for the approximation highlighted above. "Particle" is supposed to be defined relative to the vacuum state (motto: particles can be counted, and the vacuum state has none of them), so if the definition of the vacuum state is ambiguous, then so is the definition of "particle."

do particles lie in irreps of the Poincare group because Minkowski space has a global Poincare symmetry, or because it has a local Poincare symmetry?

Depends on what you mean by "because," but we can at least say this: In a QFT with a smooth background metric, we always have approximate local Poincare symmetry, so we can always use this to approximately classify particles, at least if we have a large enough gap between the curvature scale and the scale over which physically reasonable states are entangled. We can make the classification pristine when the symmetry happens to be global, but that's a special case.

Local symmetry? Gauge symmetry? Asymptotic symmetry?

Observables are invariant under gauge transformations. Is local Poincare symmetry a gauge symmetry?

Let $G$ be a group of mathematical symmetries of a given model. If it includes a subgroup $H$ under which all observables are invariant, then we refer to transformations in $H$ as gauge transformations. We don't always use the word "gauge" consistently, though, because the group $G$ may also include so-called "large gauge transformations" under which observables don't need to be invariant. (That's a choice we can make when defining the model.) The symmetry group $G$ may also include asymptotic symmetries that are continuously connected to the identity but aren't included in the gauge group $H$.

For the sake of the present question, we don't need to worry about treating diffeomorphisms as gauge transformations. We're not dealing with quantum gravity here. In QFT, the metric is a background field, and the symmetry group $G$ does not change the model's background fields. A symmetry transformation in $G$ might have the same effect as a corresponding transformation of the background field, and we can enlarge the symmetry concept so that symmetries act on families of models instead of individual models (like we do when discussing the $U(1)$ gauge symmetry of quantum mechanics with a background EM field), but for the sake of the present question, the metric field is just plain fixed.

As explained in one of the linked answers, we can use asymptotic symmetries (modulo gauge symmetries) to classify states in the Hilbert space, and if we have a reason to interpret some states as a single-particle states, then we can use that classification of states to help classify particles, just like we traditionally do in flat spacetime. Such a classification might not always agree with the approximate classification we would get using local Poincare symmetry (ante de Sitter spacetime still has approximate local Poincare symmetry!), but I don't see why that would be a problem. We can often classify the same things in different ways, and different classifications have different uses. That might sound like heresy when applied to an idea as fundamental as "particle," but that just means that our fundamento-meter is miscalibrated. In QFT, particles are phenomena, usually transient phenomena. Particles are not axioms, so they don't need to be unambiguous, much less uniquely or unambiguously classified.