What happens if you steal an Elixir of Immortality and then use it?
It goes into Bob's library (the owner of the card).

From the Comprehensive Rules:


  400.3. If an object would go to any library, graveyard, or hand other than its owner’s, it goes to its 
  owner’s corresponding zone.


There is also specific errata on the Elixir


  Elixir of Immortality - {1}
  Artifact
  {2}, {T}: You gain 5 life. Shuffle Elixir of Immortality and your
  graveyard into their owner's library.
  
  This new wording correctly instructs the player as to what to do in
  the case where the controller of the ability does not own Elixir of
  Immortality.
  
  Please pass this information on to other interested parties who may
  not have received it.