The answer, as you might've guessed from all of the above responses is, "It depends."
If you expect heavy dew or frost, a tarp will help to create a microclimate for you that limits those on your hammock. If you expect aerial bombardment of the nasty kind, it'll help prevent that.
But, if neither of those is a particular concern for you (and most times, it's not for me, and I live in an area where there're always birds and humidity), you can most definitely go topless. I prefer to leave mine set up in the Velcro keeper straps I use for it (serves the same function as snakeskins), with the stakes for it planted at pre-positioned spots, so that--should the skies decide to open up at 3:00 AM--I can have the tarp covering me in less than sixty seconds.
That being said, make your own assessment of conditions where you're sleeping. Being wet (or worse) is less than fun...