I work at an academic institution and have participated in journal
clubs. It is not uncommon for members of journal clubs to distribute
pdfs of copyrighted papers by email or photocopies to members of the
journal club. Is there an instance of a lawsuit resulting against a
journal club or one of its members for distributing copyrighted papers
amongst members of the club?

You haven't specified jurisdiction, but most (if not all) copyright
laws exclude photocopying for the purpose of learning/studying from
violation of copyright laws. In other words: it is *probably* not
against the law, but you'll have to check with a copyright lawyer to
get a definite answer.