I seem to recall from my reading (a while back) that kerning was a problem for the Gentium font in accented Latin characters. Is this still the case? And why would the kerning be different for accented Latin characters than for non-accented ones?
Kerning of accented characters is still suboptimal with a freshly downloaded version of Gentium Plus. Note the collision in the pairs fà and ïb as well as the overly large gap between ľ and e in the below example. By contrast, Linux Libertine solves these problems by contextual forms (for the f), better kerning (between ľ and e) or does not encounter them in the first place (ïb).
The example also illustrates why accented Latin characters may need a different kerning in the first place.
From my experience, most fonts lack proper kerning. As annoying as it can be, adjusting kerning is just a part of typography. No font can get it perfect right away and they usually needs some adjustment. The initial kerning of a font is whatever the font creator made it. So unless the difference is substantial or you are working with a ton of copy, I don't see a solution other than manually adjusting.
"And why would the kerning be different for accented Latin characters than for non-accented ones?"
Because they are different in shape! Kerning is all about helping all those shapes comes together in an harmonious way. Look at the well-done visual examples in Wrzlprmft's answer.
Gentium is published by SIL and I am receiving non-regular mails about updates. I do not remember reading about kerning ever, but I recommend to get on their site and sign-up for news-mails if you are interested in the improvement of certain typefaces and their fonts.
Another answer would be this: Download the freshest version (don't trust in what your OS is providing out of the box) and see for yourself.
This is assuming that you are indeed working with texts that carry accented Latin characters. Just set to all-defaults in your preferred DTP tool, paste some sample text and zoom-in muchly to look for real-live problem spots. The license of Gentium allows you to improve it for your own needs (I have hacked some SIL fonts for other issues) - one of the conditions is just that you give it a new name.
Declaration: I work with SIL but in a very different corner. So I am also a user of those fonts, not a provider.