Interesting thought....

I think all types of memory are active when we are asleep, at least for recall, because in your dreams you can remember things from different time spans.
Short term and medium term is available for storage or else you wouldn't remember them when you woke up, or you wouldn't be able to recall them later in the day.

Long term however usually take effort to establish. Usually a concious effort with repetition.

Think of the clutter after a while if dreams were stored in long term....
The only dreams I commit to long term are those that have a personal significance to me.

Guess we can't throw away those dream journals yet, eh?