Oathsworn Oracle: The Oracle will only become your Follower if you have a quest. You must discard the Oracle if you complete a quest.

Whenever you draw Adventure Cards, you may discard one card of your choice that you do not wish to encounter and draw one more card to replace it, which you must encounter.

Question: If I encounter Oathsworn Oracle and do not have a quest, does she stay on a space or is she discarded?

We play it that if you encounter the oracle and have no quest the oracle stays on the space until someone with a quest comes and takes it as a follower. The oracle is only discarded when the person completes a quest.

"discarded" meaning being placed back on the space?

I agree with lonely writer.

The Oathsworn oracle stay on the space until someone with a quest visit him

After you complete the quest, then the oracle follower is discarded

No, "discarded" as in placed in the discard pile