[ncmncm]

This assumes there is no planet or moon handy to whip around, to end up going in a completely different direction, with no extra energy expenditure.

It is tricky (but possible, with cleverness and a careful schedule) to gain or lose energy this way, but it doesn't matter. If your closest approach is well within the sun's photosphere, it doesn't matter how fast you're going when you get there. So, you can do it with essentially zero delta-v, starting and ending with the same total energy as an object would have co-orbiting with earth, but on an extremely eccentric orbit.

It's not terribly rare (on a geological timeline, at least) for comets to dispose of themselves this way.

Anyway, what is so great about dropping them in the sun? Jupiter swallows comets frequently. Mars is a squalid dump, and so is Venus, at least below the clouds.