How does Underrealm Lich interact with Sylvan Library during your draw step?
This answer assumes you have invoked the "may" clause of the Sylvan library.  If you haven't, then there's no question.

Since you have replaced your draws, with "look at top three cards ..." you are in fact not drawing cards.

After you replace all your draws, you move to the next step of Sylvan library.  "Choose two cards in your hand drawn this turn."  You don't have any cards in hand drawn this turn, so you can't choose any.  [i.e. you can't put any back on top of your library.]  So, the life loss portion doesn't apply.

Final state:  3 cards added to hand, 6 cards put in graveyard, and no life lost.

All the above, is essentially a walk through of 


  120.6c Some effects perform additional actions on a card after it’s drawn. If the draw is replaced, the additional action is not performed on any cards that are drawn as a result of that replacement effect or any subsequent replacement effects.


Keep in mind the "do as much as you can rule." of which, 120.6c is essentially an instantiation of in this case.


  609.3. If an effect attempts to do something impossible, it does only as much as possible.