Can I be killed if a Dinosaur defecates on me?

Everyone in Ark poops. And the feces are quite useful for growing crops. Because of this, I often follow a Brontosaurus to collect the brown flat cakes they leave behind.

If I get hit by a turd, could I die or lose health from it?

No, having something defecate on your character does not hurt it. Unfortunately, I found out the hard way.

Ark uses the same mechanic for all dropped items, even though eggs and feces have a different look, they function the same as regular dropped items. There are a few ways to tell nothing is happening:

• Sound; Ark is (as of writing) still in development, but the game is very detailed. The fact that you don't hear an item dropping on the floor (only the defecating itself) hints that this is not an oversight.
• Forward movement; Items and feces are dropped/thrown rather than launched, which happens with stones (slingshot), arrows and bullets.
• Combat; Would a dropped item do damage, pets in the vicinity could start a fight while there wasn't any real combat initialized and the owner is on the other end of the map.

Knowing this, it should be safe to follow any Bronto you find. Watch out for flingers though, it could get pretty messy.