Do I discard weaknesses when they are done?
Weakness (or treachery) cards go to your discard after being completed. Page 20 of the RRG:


  When a treachery card is drawn by an investigator, that investigator
  must resolve its effects. Then, place the card in its discard pile
  unless otherwise instructed by the ability.


Further, on page 7 of the RRG (emphasis mine), pertaining to weaknesses that are enemies and can be defeated:


  if an enemy has damage equal to or higher than its health, it is
  defeated and placed in the encounter discard pile (or in its owner’s
  discard pile if it is a weakness).


Non-enemy weaknesses specifically say on the card what to do with them. For example, Cover Up says: 


  When you would discover 1 or more clues at your location: Discard that many clues from Cover Up instead.
  Forced - When the game ends, if there are any clues on Cover Up: You suffer 1 mental trauma.


This means that Cover Up stays on the table (with no clues on it) until the end of the scenario.

In contrast, Daisy's weakness (the Necromonicon), says: 


  ...if The Necronomicon has no horror on it, discard it.