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.