I've been through enough different HTML variants whilst not having to adhere strictly to standards that I'm moderately fuzzy what the current state of the standard is. I hear the current standard is also ... large.[1]
But even if it's not strictly necessary to balance <p> tags ... it is necessary to do so with many other HTML elements, and missing or mis-typed tags can utterly bork a page, particularly if there's any complexity to it.
(Hand-crafting tends to minimise that complexity, but it's still possible to get reasonably twisted.)
That said, checking one of my favourite HTML5 references, whose page source itself is a beautiful example of clean HTML ... I see that Mark Pilgrim in fact omits the close tags on his paragraphs:
