Is it ethical to use someone else's research as an illustrative case?
The entire point of publication is to get your work out there for other people to use as a basis for further work. (The whole "standing on the shoulders of giants" thing.) 

If a researcher is going to be upset about someone else taking their work and using it as a basis of further work, then they shouldn't be publishing it in the first place. (E.g. many companies keep research results "trade secret", which they use internally and never publish.)

In the fields I'm familiar with, it's perfectly acceptable to take published results and re-analyze them or use them as a basis for further work.

That said, there are some caveats. Among these (but not a comprehensive list): Be sure to give appropriate citations and credit as to the origin of the work at every reasonable opportunity - avoid implying that this is your data, even by omission. Be careful about copyright - verbatim copying of text and figures without explicit permission is usually proscribed, but using the underlying data (facts, numbers, general ideas) is normally not. Be sure to conform to the particular norms in your field, if any, about re-use of datasets - in some fields there's norms about data re-use, as data collection may be laborious and researchers who spend a lot of effort making datasets don't want to be "scooped" on follow-up work by people who swoop in  with a quick re-analysis.

But generally speaking, if the data are available from a published source, you're not breaking copyright, and you're not doing anything that's going to prevent the original authors from publishing additional work, you should be able to use the published data in your paper. (With appropriate citation, of course.)