Though they could use autosomal DNA, it would be extremely difficult to prove a relationship since there are so many generations between Richard III and his 17th great nephew. They would only share a microscopic percentage of autosomal DNA (something like 0.517 = 0.000763%).
Fortunately, Richard III's 17th great nephew is a direct descendant of his sister on the maternal line, meaning that they can use mitochondrial DNA. A benefit of this is that there's lots of mtDNA in our cells, so it's more easy to extract from ancient skeletons than nuclear DNA.
I'm sure they will do a very high level test (many more markers than you can easily test through a commercial company). While it is impossible to prove that two people were related using DNA, and there certainly will be differences in their mtDNA because of the 16 intervening generations, the test should be able to show whether there is a chance that the remains are related to the descendant. As you say, DNA is very good at disproving relationships, but not so good at proving them (because there is always the possibility that sequences could match by chance).
Alexander