It would just burn up and spread through the sun, so
scientists wouldn't do it as there's no scientific benefit from it. If it was possible to send nuclear waste to the sun, it would still be cheaper to bury it in an asteroid thick enough to shield the waste from outside interference, or leave it at a Lagrange point, or in Jupiter. Even ejecting the waste from the solar system probably takes less fuel than throwing it into the Sun. (Source: many hours of
Kerbal Space Program ;) )
EDIT:
Chart of fuel costs from Earth to various points in the Solar System. Based on these numbers and without using 'gravitational slingshot' maneuvers around Venus and Jupiter to save fuel, sending something from Earth's surface to the Sun takes over 208 km/s delta-V, but ejecting an object from the Solar System takes only 18.15 km/s. That's ~11.5 times more delta-V to send waste into the Sun than to eject it from the solar system, which means a lot more than 11.5 times more fuel (you need more fuel to carry more fuel, so you get diminishing returns). Ejecting something from the Solar System into the great black beyond is a pretty sure way of making sure nobody will look for it or find it.
You'd have to be dealing with something a lot more dangerous than radioactive waste to want to destroy it so thoroughly that throwing it into a star is your only option.