Well yeah, it would permanently remove the waste from the Earth and reduce it to indisdinguishable subatomic particles and plasma inside the sun. Adding it a little at a time to a stream of rocket exhaust with the velocity to easily reach the sun is highly plausible.

Could this damage or affect the sun somehow? No, every week the sun spits out about 1 to 20 coronal mass ejections of roughly a billion tons each. All the nuclear waste in the world amounts to about 250,000 tons, and the rocket would send maybe 100 pounds at a time toward the sun. Very tiny farts compared to a continuous series of very large hurricanes.