Short answer: No.
Long answer: The tarp is set at whatever angle by you when you hang it. If it's nice weather, you tie it closer to horizontal to let the breeze in and see around you. If you know rain is blowing in, you stake the tarp down like a pup tent.
In a forest, you don't get the free space to have wind blow rain sideways like you would in the open, and even if it did, it'd have to come under the longest axis of the tarp, and your hammock is set back from those ends.
It's nice and dry and depending on your hammock, it might even pack up easier than a tent after a rain. (just disconnect the rain fly and pack the dry hammock up, or if you have a hennesy, unhook the fly and roll those snake skins down, etc...)
I've never woken up wet in a hammock. We've probably all woken up wet in a tent.