.. that after a vampire (or any other unique minion) has been burned, and a new copy of the same vampire has been brought into play, the new vampire is not affected by cards that have been pointing to the original vampire? For example, Lodin enjoyed the blood flow from The Rack, and is then burned in combat or by diablerie. In the next turn a new copy of Lodin is moved to the ready region. This new copy of Lodin does not receive the benefits of The Rack (even if it hasn’t moved yet to point to another vampire).
For all intents and purposes, different copies of the same vampire are considered different (except for contesting). Another example is that even if the first copy of has yielded his title (e.g Lodin has not been the Prince of Chicago anymore), the new copy of the vampire, still has the title (and would contest it again, if there’s another vampire holding the same title in play).
This, of course, is not really relevant for cards that have been placed on a vampire directly. Cards like Fame, or Heroic Might go the ash heap as soon as the vampire attached to the cards is burned or diablerized.