The ListView of documents  can certainly only show listitems that meet a
criteria like [Today] is less than ExpireDate.

How do I set  the default view
into the listview in the "Shared documents" page?
I am quite new to SP. Your detail explanation is truly appreciated.

1. Click on Documents and Lists in the top nav.
2. Click on Shared Documents (or whatever your document  library is
called).
3. Click on Modify settings and columns at the bottom of the left nav.
The Views section at the bottom of the resulting page can be used to
manage the ListViews for your document library.
4. Click on the All Documents ListView (or click on Create a new view).
5. Select the fields you want to see, sort them, filter them, etc.
Specifically, choose the ExpireDate field in the Filter section and then
set criteria like "is less than" and provide [Today] as the column
value. If you don't have an ExpireDate you will need to create a date
field on the previous screen in the Columns section and make sure it is
a required field. Document libraries are harder to enforce required
metadata for a document.
6. Click the OK button.
7. Near the top of the page, click the link to "Go back to [Your List]".
8. Choose the All Documents or your custom ListView from the left nav.