If you've ever wanted to change "File System Path" in Drupal 6 after uploading files to your site, you'll know that it can be tricky. You are warned that
Changing this location will modify all download paths and may cause unexpected problems on an existing site.
And it does. The most likely symptom is that your images or downloads will no longer work.
This is because Drupal stores the file location in its database at the time the file is created, but it uses this stored value in conjunction with the File System Path to work out where the file is stored whenever a user requests it. After a change, the two locations no longer match up and Drupal gets confused, meaning that your images and files no longer work.
SQL to the rescue
It's relatively easy to fix this, assuming you have access to your database.
NB: If you haven't yet changed the File System path take a backup of your site. You should be doing this regularly anyway and especially when making major changes.
1. Check that this is the fix for you
Firstly, confirm which files you'll be updating by looking for entries in the database that have the old File System Path stored ('<oldfilepath>')
SELECT filepath from files where filepath like '%<oldfilepath>%'
e.g. if your old File System Path was set to 'sites/all/files' you'd use
SELECT filepath from files where filepath like '%sites/all/files%'
The % at the start and end is a wildcard in MySQL and will match any filepaths that contain the string sites/all/files. If you don't see any files listed, check your query and if you're sure its correct you need to look elsewhere for a solution.
2. Make the change
Assuming the query returns some results, you can now update those records with the new filepath. e.g. lets say you're moving from 'sites/all/files' to 'sites/website.org/files'
UPDATE files SET filepath=REPLACE(filepath,'sites/all/files','sites/website.org/files')
3. Clear your cache and test
Clear the Drupal cache and check that your images are now displaying OK.

What about inline images?
Thanks for the precise information. I had to change today the filepath of a site in a multisite environnement and followed your hints. All worked well, but the inline-images in node bodies inserted by IMCE.
I had to manually change all the image properties :-( I think there doesn't exist another solution, since the filepaths for inline-images lies in the html-img tag in the node body. Or does anybody knows an automated solution?
Thanks!
Hans-Peter
Scanner module might help?
You could try the Scanner module which claims to be able to find and replace across all CCK fields. You could then do a find for the old address and replace with the new one.
Or perhaps the Pathlogic filter which can do this transparently for you, at the time the node is viewed?
I've not tried either, but I'd be interested to hear from anyone who has.
Nik