The only way you could do the lamb harm by picking it up (aside from dropping it, or otherwise injuring it by handling it wrong) is if for one reason or another it wasn't accepted back into its herd, or if your interaction gave it a reason to leave the herd.
Some animals will reject their dependant young if they smell like human, as far as I'm aware this is not true of sheep. Domesticated animals don't reject their offspring if they detect human scent. It's a known behaviour for some deer–which is why petting newborn fawns is taboo–but often times this largely depends on the mother, as it's not guaranteed that all does will react the same way to the smell of human on their young. It also depends on the size of the baby, it's only at risk of abandonment if it's too little to keep up with it's mother, or if it hasn't weaned yet.
Sheep are often handled by people (sheering, herding, children taking lambs as pets, etc.), I don't think they're going to feel threatened if they catch a hint of one. I think what that other hiker was referring to is the potential the lamb could have to taking a liking to you to the point where it might leave it's herd and follow you home, or come looking for you and end up lost and alone. Lambs are a lot like puppies in that they can crave affection. If you give it affection (picking it up and holding it) you could win the lamb over and make it want to come back for more.