The Old Dream tile actually repeats the Destiny tile as it was experienced. It also repeats the Change, Chest, and Monster tiles as they were experienced. It won't ask you the same questions nor give you the same number in Guess the Number. Not sure about how it behaves with regard to the Teleport tile.
EDIT: Using the Old Dream efficiently is key to scoring 700-800 scores.
For teleport it send you back to the same tile that the teleport originally sent you to.
The opinions expressed on this post are my own as a player, not as a Moderator.