If gravitational waves are influenced by the deformation of spacetime around a very dense object like a black hole, and thus can be "reflected" back to (the future position of) their source, doesn't this mean that an object can be graviationally affected by its past?
Can this become a non-negligible effect in some situations? (Presumably at very close distances, due to the combined inverse r^4 falloff)
The sort answer is: Yes.
To explain in detail it is useful to look at the notion of Green's functions for the gravitational field. Suppose that the object we have is small enough that we can treat the gravitational field (metric) it produces as a perturbation of the gravitational field of its surroundings. We can thus write the gravitational field as
$$ g^{\rm full} = g^{\rm background} + h$$
(For simplicity we are not writing any of the tensor indices.) The metric perturbation $h$ will satisfy the linearized Einstein equation. Like other linear partial differential equations we can write the solution in terms of a Greens function $G(x,x')$
$$ h(x) = \int G(x,x') T(x')dx',$$
where $T(x')$ is the energy-momentum tensor of the object, and $G(x,x')$ is the field produced at event $x$ sourced by a delta function at event $x'$.
If the background spacetime is ordinary flat Minkowski space than $G(x,x')$ is only non-zero on the past light-cone of $x$ (This is Huygens principle). And since the wordline of a massive object in Minkowski space can never cross its own lightcone, an object cannot feel its own gravitational field (including any gravitational waves produced).
This all changes when the background is curved. The first change is that in curved background, Huygens principle will fail. Gravitational waves continually get refracted, and consequently, the Greens function $G(x,x')$ at $x$ becomes non-zero for points in the timelike past of $x$. This means that the gravitational field at the object becomes a function of the past history of the object. This field also inlfuences the motion of the object, showing up as an effective force term the motion of the center of mass. This is the so-called gravitational self-force (or GSF).
Now, while the GSF of an object formally depends on the full history of the object, it usually is dominated by the object's most recent history. This can change because of the second difference of how things work in curved spacetime. In curved spacetime it is possible for the past worldline of an object to cross its own lightcone (e.g. because the lightcone has wrapped around the lightring of a black hole). This can introduce strong(er) interactions with object's more distant past. This is probably the closest to what the OP had in mind with their question. In explicit example of this happening was examined in this paper, which studied the quasinormal modes of a black hole excited by another object passing on a highly eccentric trajectory, and their impact on the object's GSF.
