Remove card from game go in trash in Dead of winter?
No, every card that you would put in the waste pule says for it to be placed in the waste pile.

A good example of this is resolving a crisis as you don't add those cards to the waste pile when you are done. Here is the online rules


  Resolve Crisis: Shuffle the cards that were added face-down
  to the crisis during the player turns phase. Reveal them
  one at a time. Each item card added that has a symbol
  matching a symbol in the prevent section of the crisis
  is worth 1 point. Each card that doesn’t have a symbol
  matching a symbol in the prevent section of the crisis
  subtracts 1 point. After revealing all of the cards, if
  the combined point total is lower than the number
  of players then immediately resolve the crisis. If the
  point total equals or exceeds the number of players
  the crisis is prevented. Additionally, if the point total
  exceeds the number of players by 2 or more the colony
  gains 1 morale. After resolving the crisis, remove all
  cards added to the crisis from the game. See Example:
  Resolving a Crisis to lef