When discussing things in general relativity, one generally strives to describe physical phenomena in a way independent of arbitrary coordinate choices, so that one can be sure they're extracting real physical information about the model rather than coordinate artifacts. Forces on point masses can be described in a relativistic setting in keeping with this philosophy through the concept of four-force, the covariant derivative along the mass $m$'s trajectory of its four-momentum $P^\mu = m U^\mu$, where $U^\mu$ is its four-velocity:
$$ f^\mu = U^\nu \nabla_\nu P^\mu.$$
Covariant descriptions of classical electromagnetism describe the Lorentz force on a charged particle as a four-force, for example:
$$f^\mu_\text{Lorentz} = q F^{\mu \nu}U_\nu,$$
where $F^{\mu \nu}$ is the electromagnetic tensor and $q$ the particle's charge. The magnitude of the net four-force is physical in the sense that it is the force the particle "feels" in its rest frame, i.e. its rest mass times its acceleration according to a momentarily co-moving inertial observer.
In general relativity, it is posited that test masses under the influence of gravity alone follow geodesics, for which the equation of motion is
$$ U^\nu \nabla_\nu U^\mu = 0.$$
Up to a factor of the constant mass, this is the statement that the covariant derivative of the particle's four-momentum is zero along its trajectory, i.e. that the net four-force is zero. This is what is meant when people say "gravity is not actually a force"-- the action of gravity alone is that of zero four-force. Though test particles following this motion can be said to have varying coordinate accelerations in different coordinate systems, they are not accelerating in the only invariant sense of acceleration available. They are following the straightest possible paths through the curved spacetime manifold.
So, what about work? Can we achieve a similarly natural relativistically invariant analog? Well, in a sense, though the concept doesn't seem to be discussed or utilized very much. Let's relax the assumption that our particle has constant mass, and recall that the four-velocity is the tangent vector to the trajectory when parameterized by proper time $\tau$, so that $U^\mu U_\mu = -1$ (I like signature $(- \, + \, + \, +)$ and units with $c = 1$), and we have
$$-2m \frac{dm}{d \tau} =\frac{d}{d \tau}( -m^2) = \frac{d}{d \tau} (P^\mu P_\mu) = U^\nu \nabla_\nu (P^\mu P_\mu) = 2 P^\mu U^\nu \nabla_\nu P_\mu = 2 P^\mu f_\mu$$
This then reads,
$$\frac{dm}{ d \tau} = -U^\mu f_\mu.$$
The left hand side is the rate of change of the particle's rest mass with proper time, while the right hand side is what one would obtain by taking the Newtonian expression for work done per unit time (or power), $P = \vec F \cdot \vec v$ , and minimally substituting in the invariant relativistic analogs (the sign is an artifact of the signature-- if the vectors are parallel, the quantity is positive). This says something like "the relativistic work done by the net four-force is the change in rest mass". This might be what you'd guess, considering that the rest mass is the only coordinate-invariant measure of energy available (kinetic energy is right out, of course), and work is supposed to describe a change in energy.
In this sense, gravity of course does no work, as it contributes no four force. The Lorentz four force listed above also does no such work, since the tensor $F^{\mu \nu}$ is antisymmetric and hence $F^{\mu \nu} U_\mu U_\nu = 0$. This is saying, e.g., that electrons don't gain rest mass as they accelerate under the electromagnetic field.
That's the invariant discussion. Things get a lot hairier when attempting to use coordinate-dependent notions in GR. One can attempt to have this discussion in a coordinate-dependent way by considering the coordinate acceleration associated to geodesic motion, what one might call the acceleration due to gravity. In local inertial coordinates (i.e. a coordinate system "almost" special relativistic / Minkowski in a small region), at least, one can approximately recover the usual kinematic sense in which this coordinate acceleration does work in accordance with the work-energy theorem.
In a general coordinate system, though, there's not really a meaningful sense in which this is the case. There's difficulty defining what the terms should mean, so it depends on what you want to mean. When the coordinates are structured well enough that one can meaningfully consider a notion of coordinate kinetic energy, it will generally change under geodesic motion (so gravity does "work" in the sense that it impacts the coordinate kinetic energy), but it is not nicely related to any of the natural candidates for what one might mean by the coordinate force doing work as described through power: $\frac{d}{dt}(P^i)v^i$, $\frac{d}{d \tau}(P^i)v^i$, $\frac{d}{dt}(P^i)v^j g_{ij}$, $\frac{d}{dt}(g_{ij} P^i)v^j$, $m\frac{d}{dt}(v^i)v^i$, $m\frac{d}{dt}(v^i)v^j g_{ij}$, etc. (here $\vec v$ is the coordinate velocity and $P^\mu$ the four-momentum). In general, of course, there need not even be a nice break down into $3$ spatial coordinates and a time coordinate, so that none of the prior expressions even make sense. The best one can do with these expressions is say that, in coordinates where they make sense, these coordinate quantities that look like work are nonzero but have no consistent relation to actual energy.
What's the takeaway from all of that? It's that, yes, if you fully translate to the relativistic coordinate-invariant language in which gravity is not a force, gravity indeed does no work. However, work is typically not discussed in GR at all, so it's kind of a conceptual mismatch. The coordinate-invariant notion maybe isn't the most useful, and the coordinate-dependent notions either don't make sense or are, at best, strictly qualitative in their analog to the Newtonian situation. Only in special relativistic limits can one be sure the relations like the work-energy theorem are recovered, so that quantitative work of the kind that gravity would do as a coordinate force is really a special relativistic concept.
