If it isn't stated to place cards on the discard pile do you leave them on the space?

What kind of card are we talking? Most cards that stay on the space tell you to leave it on the space. Stranger's and Places are typically cards that are left on spaces.

IF, a card is not encountered due to a Character having a condition happen in which they do not encounter all cards the ones not encountered remain on the space to be encountered by the next adventurer to land on the space. If a character dies or becomes a toad, ALL of his cards are dropped on that space.