<p>I have just started migrating my homegrown persistence framework to JPA.</p>

<p>Given that the persistence frameworks hide a lot of the plumbing, I'm interested in knowing if NOT closing EntityManagers will create a resource leak, or if the frameworks will collect and close them for me.</p>

<p>I intend in all places to close them, but do I HAVE to?</p>

<p>At the moment using TopLink, just because it works with NetBeans easily, but am happy to investigate other JPA providers.</p>

You should.
Frameworks have no idea how you intend to use the EM, so they cannot close it (except, may be, on finalization, which is not guaranteed). Yes, not closing them would create a resource leak.
The idea is the same as "always close java.sql.Connection" (despite some data sources have settings to close them automatically by inactivity) or "always close Hibernate session".
Besides, if you plan to write portable code, you shouldn't rely on specific JPA provider "being smart" -- other may fail to close the EM in time.