Identity maps are used to implement the first level cache in many Object-Oriented Mappers. If your application stack involves Entity Framework or nHibernate (or yet another orm) most probably you already have an identity map there. But this is what you probably know already.
The question whether or not an identity map could be static doesn't have a definite answer and I could probably imagine a situation where it could work but drawbacks are severe:
- the cache would not be able to easily see any external changes
- the memory consumption would grow without control
- concurrency issues could occur
Implementation of an identity map which would be free of all these issues would probably be unnecessary difficult. It is just safer and less expensive to restrict the map's lifetime to a single request. Note that it still does its job - if a request involves multiple reads, the identity map serves the data rather than the database when applicable.