From what I've read people have had good experiences using false bottoms with leaf hops, and the hops actually act as a filter for the break material. I've seen varying opinions on false bottoms with pellet hops. If you normally brew with leaf I'd go false bottom and be sure to use bags anytime you are using pellets.
I'd have a false bottom in my keg, but I use an immersion chiller and I'm not sure how it would fit in there with a center dip tube. And I can't drill low enough to put the pickup tube below the false bottom.