Besides the fact that would be incredibly expensive, even after you've made it to space, getting to the sun is really, really hard. You essentially have to cancel out most of the Earth's orbital velocity.

We produce millions of tons of trash daily, and it requires an insane amount of rocket fuel to get things to space. Saturn V requires 6 million lbs of fuel to carry only 280,000 lbs of cargo. 21 lbs of fuel for every 1 lb of cargo. And that's only to Earth orbit. Getting to the Sun is much further. We'd be out of fuel in day one...