Will sticky mess dropped onto hostas and other plants die

If you have ever had that stuff drop onto your car, you will know that hosing off with plain water is not too effective.

You might want to check the tree if its not too large to treat - likely its got an infestation causing the sap to drip onto plants below. The most likely contenders are aphids or scale insect, but if the tree is too large to spray, then there's not much you can do about it. The 'sticky stuff' shouldn't kill the plants beneath, though you can try spraying them daily with a hose to try to keep the stuff from hardening. Once its set and hardened, its difficult to remove. It should ease off towards the end of July, when the sap's not causing so much turgidity in the branches and leaves.