Index theorem of Atiyah and Singer closed a substantial field of research in the 1960s. I knew people who were working in this field, and had to switch the field of their research completely.
A more modern example is Louis de Branges proof of the Bieberbach conjecture. There was a large field of research, I would say a central field in analytic functions theory, which could be called "coefficients estimates". To be sure, it still exists, but nowadays it is considered marginal. Contrary to all expectations, the highly original proof of de Branges's theorem did not lead to a significant further development (so far).
Another commonly mentioned example is Hilbert's results in the theory of invariants. They closed the field in some sense, though not forever.
It also happens sometimes that a new breakthrough does not really close the field, but many people have to switch to another field because they are not equipped to understand the breakthrough. I do not want to give modern examples of such a sad situation, but according to Lev Pontryagin's own published recollections, he switched from topology to applied analysis in 1940s because the new abstract language introduced by the French revolutionized the area, and he could not stay in line with the modern development. (Pontryagin was one of the most prominent topologists of his time, and he was 32 years old in 1940.)
