Has any research been done on the effect of introversion on language learning?

This is a very hard area to research because of so many confounding variables and the fundamental difficulty of measurement of both introversion (or similar personality traits) as well as progress in learning (both reliability and validity are problematic in most measures).

However, there's been no end of research on personality and SLA (second language acquisition). Most of it has used a version of MBTI (Myers-Briggs). But given all the difficulties, there is no good reliable evidence of a strong correlation between personality traits and learning success. One suggestions made was that personality interacts with other factors (such as learning context) but is not in itself a predictor of learning outcomes. A good survey of the literature is in this book by Zoltán Dörnyei.

However, it's not at all clear how this should be applied in practice. The most convincing case for using personality as a part of the mix in instructional design with a clear methodology was this book by Betty Lou Leaver.