I was thinking sed, too, but to replicate the behaviour (and scene):

echo "HOLD THE DOOR" | sed -n '
  p
  s/E//g p
  s/L//g p
  s/TH//g p
  s/ //g p
  s/DO//g p'

Note: -n suppresses the default printing that sed does, so I could include the last p for symmetry and clarity 🙂