Probably the App Pool would be restarting. This could happen every couple of minutes if there is a shared server with several sites sharing an app pool and the app pool is set to recycle if it uses up too much memory.
The only fixes for this are:
a) move onto a different hosting environment
b) use cookies to identify the user and look them up in the database (eg store an encrypted user ID in a cookie and store any related data in the database)
If you have a lot of other stuff in session memory you could consider implementing a database table just for storing session state (in classic asp you would have to roll your own - in .net this is a standard config option).
If it is a big app with a lot of reliance of session variables you would want to go for option A if possible.