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Nuclear waste is extremely dense and therefore extremely heavy. It would cost an incredible amount of money to launch it into space, let alone the sun. The corollary is it's also really valuable since it still contains an incredible amount of energy, our current reactors only burn 5% of it where as future reactors will be able to burn all of it. An analogy I often use is we're burning just the kindling and our 'waste' is the hard oak that's harder to burn.

That question is asked and answered here:

http://www.reddit.com/r/askscience/comments/2r43nl/why_dont_...

Any rocket that could deliver nuclear waste to the sun would use an awful lot of fuel, given the density of the waste, and would be a very big problem if there was a failure shortly after launch.

Also, anything that could deliver radioactive waste to the sun could deliver it to a major city just as easily.

The risk of fallout and the fuel costs are too significant.

Easier to build a modern nuclear reactor that can use nuclear waste for fuel- there are projects in the works to do just this.

What do you do when the rocket explodes on launch?