It's actually easier to leave the solar system than it is to hit the sun.
(Earth orbits the sun at 30 km/s [1]. That means you need -30 km/s of ∆v to kill the energy we're born with. The escape velocity for our solar system from Earth's orbit, meanwhile, is about 42 km/s [2]. So you just need 12 km/s of ∆v to skip town.)
[2] sqrt((2 * 6.7E-11 * 2E30) / (150 * 10^6)) given G~6.7E11 [a], mass of the sun is about 2E30 kg [b] and the Earth orbiting the Sun from about 150 million km [b]; for escape velocity [c]
That's good calculation there, but the number we're concerned with is the amount of ∆v that the spacecraft has to generate itself. If you just get close enough to the sun, a spacecraft could use the solar wind and solar magnetic field to shed more velocity, until it is close enough to aerobrake in the sun's atmosphere.
Once you're out in the Oort cloud on the way out, you can't exactly deploy a solar sail to get another push (unless you make it impractically large).
The outermost layer of the Sun ends at 0.1 AU [1]. To put that in perspective, Mercury orbits at between 0.3 and 0.5 AU [2]. (It costs about 13 km / s to get to Mercury exiting from LEO.)
It's much cheaper to fling things out of the Solar System than into the Sun.
Again, you are thinking in terms of propellant carried up from Earth.
Thrust available to a spacecraft using a solar sail is a function of distance to the sun. The sail is a fixed cost, but it may be presumed to degrade over time. It has no propellant that costs additional money to launch.
If you go in, the ever decreasing efficiency of the sail is offset by the greater available energy from the solar wind, and you may still be able to complete the mission (more slowly) with a damaged sail. Your available thrust increases with every kilometer closer to the sun.
If you go out, the decreasing efficiency of the sail compounds with the lesser total energy available to the sail, and if the sail is damaged, you may never reach escape velocity at all. You will never have more thrust than the instant the sail deploys.
That is hard (there was an article on HN about exactly this a while ago). You need a large amount of energy to cancel out our orbit or you just end up sending it into its own solar orbit (which will intersect with ours, to boot).
I don't think you understand how much energy would actually be required to launch the waste into the sun. Costs would have to be reduced by a couple orders of magnitude to make launching any significant amount of material into the sun viable.
