It is completely theoretically possible to dump things in to the sun, given enough energy. How much energy would it require? Well, there isn't a table I've seen that shows how much delta V is required to shoot to the Sun, but there is a convenient one for Mercury, which we can safely assume is less than would be required to get to the Sun. The Delta V requirements to directly get to Mercury from Earth are 48 km/s. That includes landing and returning to Earth, but even that would only add a factor of 2 or so. Given about 10 km/s to orbit, that's a huge rocket, just to get to Mercury. Real missions save some of this by doing flybys, but they require active course corrections for some time to get things just right. That's a huge amount, far more than any ship is likely to have!
So, what are some alternatives? There are basically 2 main goals for space junk removal, depending on where the item is. The first is to make it crash in to Earth (Or another suitable object). The second is to get it outside of a usable orbit.
Okay, so how are these accomplished? The first can be done by simply slowing the object down to the point where it lightly touches Earth's atmosphere at it's minimum point. Given enough time, that will cause the object to re-enter Earth. Assuming it's not that large, this shouldn't pose any danger at all.
The second possibility is to have it enter a useless orbit. GEO would require substantial fuel to re-enter Earth. Instead, what they do is raise the orbit by a bit such that it will never intersect the GEO orbit, at least, not for a very long time. This basically eliminates the possibility of impacts.