When I went to college, an old-schooled teacher of mine insisted that, because I was interested in chemistry, I needed to work on my German as it was the lingua franca of that trade. It turns out, there is absolutely no need to speak German anymore to be a successful researcher in chemistry.
I somewhat thought that, since at least two decades, English was the only language required to be a researcher in any field (excepting, maybe, literature). However, seeing this question made it apparent that some graduate programs still have requirements for other languages. How useful are they?
In a word: are there some fields where reading/writing/speaking a language other than English is necessary to perform research at an international level?
(Let's exclude fields related to literature and linguistics. Let's also forget about teaching, and focus the question on research.)
Answer
Are there some fields where speaking a language other than English is necessary to perform research at an international level?
It depends on what you mean by "speaking" and "necessary". In mathematics, there is no need to speak any language but English and the language of the country you are working in, but there's a real benefit to being able to read other languages. It's not strictly necessary, and its importance varies between fields. In my case, I don't use other languages on a daily or even weekly basis, but probably read French on a monthly basis and German a little less frequently.
In my view, the reason for language exams isn't that these languages are required to be a researcher, but rather that they are helpful. The best argument I've heard for requiring language exams is as follows. Learning how to read mathematics in French or German is not very difficult, but it's still a burden if you just want to read one paper. Usually you can get around it by finding another exposition, finding someone to explain the paper to you, or just deciding the paper is not relevant enough to bother with. However, even though this is a rational solution to a one-time dilemma, it's a bad idea if you are going to be faced with these dilemmas repeatedly. At that point, you'll gain a lot in flexibility and convenience if you can read the papers you want to read. The purpose of language exams is to keep people from repeatedly making short-sighted choices.
I'd guess that language skills are more important in mathematics than in most scientific fields, because the mathematics literature has an enormously longer half-life. In my work, I frequently use and refer to papers that are 50 years old, and sometimes quite a bit more. These results have not always made it into textbooks, and even when they have the original papers sometimes contain valuable insights.
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