When a asp.net web page is accessed remotely by a user, IIS process the request and sends the output to the user. For processing the request, it may create some temprory variables based on the program logic that you have written. The temprory memory data will be cleared by .NET CLR once the response is sent to the client browser.
IIS will not store any data in memory unless you store data specific to users in Session. If you want to save memory, do not store values in Session, Application and Cache objects.
If you want to share the instance of a class in .NET, you need to implement Single Desig Pattern.
Check the following links.
Implementing Singleton in C#