Are there aspects of General Relativity that have yet to be tested?
A good general resource on this kind of thing is the review article by Will, The Confrontation between General Relativity and Experiment, https://arxiv.org/abs/1403.7377 . He updates it every few years. The most recent version predates the direct detection of gravitational waves (although they were already verified observationally based on systems like the Hulse-Taylor pulsar [PSR B1913+16]).
If objects like Sag A* are correctly modeled as Kerr (i.e., spinning) black holes, then we have not yet observed whether they have event horizons, as predicted by GR. As safesphere noted in a comment, this may happen in the fairly near future.
GR incorporates the equivalence principle, and therefore it predicts that nothing special happens at an event horizon in terms of the local properties of space -- no firewalls or anything else crazy. This has not been tested.
There is quite a bit of work by theorists like Joshi suggesting that astrophysical collapse might not actually lead to a black hole but rather to some other object such as a timelike singularity. That is, the cosmic censorship conjecture is looking weaker and weaker. This has not been tested.
If it turns out that people like Joshi are right in their suspicions, then we could potentially observe singularities (because they wouldn't be hidden behind event horizons). This would allow us to test a prediction of GR, which is that singularities should be a generic thing that happens for most initial conditions. If we don't ever get such an opportunity, then the only singularity we'll ever have a chance to observe is the big bang.
The Lense–Thirring precession, often referred to as "frame-dragging", has not been convincingly tested. The Gravity Probe B satellite experiment was supposed to do that but it was plagued by unexpectedly large torques acting on the gyroscopes, caused by the interaction between imperfections on the surface of the rotors and their housing. The Gravity Probe B team eventually managed to model that away and announce the detection of frame-dragging with a precision of 20%. But it seems to me that the dominant opinion in the scientific community is that a repeat of this experiment is needed.