The "Unlock Login Keyring" popup is useful, but how do I make it forget a password for a specific website?
You can use seahorse to change and delete keyrings and items in keyrings.
The passphrase is stored in your login keyring. You can access that with seahorse (passwords and keys from the dash)
On the left side select login, then browse on the right side to find the entry (the description should be self-explanatory, it contains the name of the resource)
Right-click on the entry, select delete.
When you try to access the resource again you get the same dialog as before.