It always amazes me at how most people can simply not comprehend the scale of astronomical distances. The Sun is very very very far away, so far that it takes light from the Sun, 8 minutes to reach us. Yes, it's our nearest star and yes it's possible to get there but the effort is enormous, why would we put that much effort for waste. Any such attempt will likely cost us more in energy than any reasonable expectations of benefits.
Also, most Nuclear reactors are actually designed as part of a weapons program hence generating very radioactive by-products. There are very few truly civilian nuclear programs and the reactors designed from those programs don't have the same environmental issues. The CANDU reactor from Canada can actually operate on natural uranium (designed to do so in event of a disruption of enriched uranium supplies due to politics) and the spent fuel being less radioactive than the natural uranium does not require special containment. The CANDU reactor can also run on the spent fuel rods of other reactor designs, a good indication of how wasteful the other designs really are. Of course, that's not to say that the CANDU can't produce weapons grade plutonium, India delayed the construction of their CANDU by insisting on various modifications throughout the construction process, as it turns out, those modifications were to enable the production of weapons grade plutonium hence leading to their atomic bomb. There isn't any reason why the production of nuclear waste can't be addressed here on Earth provided we can keep the military requirements out of it.
Besides, why would we want to chunk millions of tons of Earth into the Sun. It would be a one way trip, and we may need that carbon or uranium in the future. Carbon is a very useful element and is the basis of life. It would be better to take the energy involved in blasting it into the Sun and convert it back into something useful such as hydrocarbon fuels or carbon nanotubes.
If you think that rocketing waste into the Sun is a good idea, then why don't you come to my place, pick up my trash and carry it on foot to a landfill in China. Don't forget your swimming trunks, the Pacific will be quite the swim.