Two JPA-specific reasons come to mind:
EntityManager is not guaranteed to be threadsafe by the JPA spec. Therefore, portable JPA apps can only use an EM in at most one thread at a time. The idiom of creating a method-local EM and closing it before it goes out of scope encourages stack confinement of EM references.
An EM using the "Extended" Persistence Context Lifetime maintains a single persistence context for its entire existence. This means entities stop automatically detaching on commit(). Instead they must be manually detached, or else the EM remains responsible for tracking them.
This question is really a JPA-specific version of the old "when to pool objects" question. That's a tough one, but the answer is probably "rarely".
This old developerWorks post by Java concurrency expert Brian Goetz expounds the point. The gist: pools make great sense for costly objects, like database connections. But for short-lived, small, and quick-initializing objects like EntityManager, pooling or some other form of long-term reference retention is a hard sell.
But, it's a general question, so there are bound to be exceptions. Maybe the application is simple or singlethreaded. Then these concerns about threadsafety become moot.