First off, if this is an ASP question, it should be posted in an ASP forum, not ASP.NET.
You will have to change the user the virtual directory that hosts the ASP app is running under to a network account that has privileges on the share you wish to connect to. Alternatively, there is a workaround based on a window "feature". You can create a local machine account on each machine involved in the connection (the web server and the file server). If that local machine has the same username and password you connect between them without additional authentication. I have used this technique to access file shares for Visual Source Safe repository access.
Generally though, getting a low privileged account designed to run web applications to have enough rights to modify network resources can be tricky and a bit of a security hole. If the application is somehow compromised you could be putting a lot more at risk than just the web app server.