Even in theory it would be difficult. With gravity alone I only see it would be possible if either there's a massive object that magically could turn on and off it's gravitational pull in sync with the earth's orbit around the sun, or if an object actually passes through the solar system just past the sun and at that moment is lucky enough to dart the earth into the sun (which would be one extreme unlikely hole-in-one).

I think there's a lot of thinking that's a bit misunderstood here, but the entire universe except for mercury and venus is actually pulling the earth away from it's orbit around the sun. In our local solar system, the sun is massive enough that's it's only a few centimeters per year that we're creeping away, but there's no way anything outside our orbit could pull us into the sun, except maybe for some bizarre theoretical phenomena, assuming gravitation is the only external force (a collision, for instance, is a completely different story!).

If a force was strong enough to do that, it would pull the sun out of that collision course as well and we'd just end up with an even more extreme orbit around our yellow friend, if we would even be able to maintain that as we accelerated towards whatever is creating the massive pull.

Yeah both would get pulled. Mass ratio between the objects has effect, but in this case both the sun's and earth's mass would be negligible compared to the black hole, so we can assume the black hole is not pulled back into an orbit around the sun-earth system.

Distance is a more important metric, as the effect is divided by the distance squared. So the further away two objects are, the less they pull each other.

Now assuming both earth and sun gets pulled what would happen is the earth's orbit would become more and more elliptical over time and the average radius would also grow over time, moving us on average further away from the sun. At the closest extreme of the elliptical path it might be possible we could pass closer to the sun than before the black hole starting pulling, but this requires a bit more mathematics than I'm willing to do as I'm supposed to be working atm.

Long story short: only the sun itself can pull us closer to the sun, but as long as we are orbiting through approximate vacuum that's not gonna happen. With friction though, say if the sun's orbit passed through a gas cloud that might happen, but we'd probably have much more serious problems before crashing into the sun if that ever happened.

Wouldn't even be possible. The gravitational pull would be stronger when we're closer to it than the sun, and weaker when we're on the far side of the sun. The earth spends half a year with it's center of mass closer to the black hole than the center of mass is to the black hole and half a year further.

Meaning if it were able to pull the earth at all, it would necessarily pull us further from the sun, not closer. The same is true for all objects outside the earth's orbit around the sun.