I upload a file and directly load the input stream into one of my objects. My question is: should we close and reset it as soon as we finish processing it?
You do not need to explicitly close it, resources allocated on your server are deleted when the request ends. See Final Note in MSDN docs.
http://msdn.microsoft.com/en-us/library/system.web.httppostedfile.aspx
However, if the question is that you should close it (or at least get rid of it), then I would say yes. Why not? It can release resources earlier than they would otherwise be released, and you know that you no longer need them.
All objects that extend System.IO.Stream objects implement IDisposable. It would be best to put your input stream in a using block to make sure it is properly closed and removed even if an exception is thrown.
using