I would like to be able to update Firefox to the latest version withing Sandboxie, without affecting my browser installation outside of it. The Firefox Tips page states:
In the default configuration, any updates to Firefox or its add-ons will happen only within the sandbox. When the sandbox is deleted, all such updates will be deleted as well.

This is exactly what I want. However, while I am able to download the update within Sandboxie, the update fails when I restart Firefox.

How can I update Firefox within Sandboxie?

[Edit] Read the Edit in red that's located in my post, in the link listed below. It was necessary for me to delete a WriteFilePath setting for the 'Program Files' folder, before I used the "Check For Updates" feature for Firefox in the sandbox.

What you do is to install Firefox inside of a sandbox, so it can be updated there and not affect the install outside of the sandbox.

1. Create the sandbox if it doesn't already exist
2. If the sandbox does already exist, make sure to use "Delete Contents" before installing Firefox in it
3. See this thread in the forum:
http://sandboxie.com/phpbb/viewtopic.php?t=12690&highlight=writefilepath
[Edit]Read the Edit in my post at the above link[/Edit]
My first and second posts in this thread describe how to use the WriteFilePath setting for a sandbox, so that your existing install of Firefox will not be seen by the sandbox install. Otherwise, the install in the sandbox will copy things from the install outside of the sandbox, and it will appear that you have the same extensions already installed, for example.

Unfortunately, that thread shows the folder settings to use, but it does not show how to get those settings using Sandbox Settings.

The settings in my first post show what folder paths should be used for XP, and my second post deals with paths on Vista/Win 7. Since you didn't say what Windows version you are using, I'll use the Win 7 paths in an example.

C:\Program Files (x86)\Mozilla Firefox
C:\Users\(username)\AppData\Roaming\Mozilla\Firefox
C:\Users\(username)\AppData\Local\Mozilla\Firefox

Sandbox Settings > Resource Access > File Access > Write-Only Access
(The "Add Program" button will not be used here)

"Add" button: Navigate to "C:\Program Files (x86)\Mozilla Firefox" and select that folder (OK in the small window)
[Edit] This may need to be removed after the initial install of Firefox in the sandbox

"Add" button: Navigate to "C:\Users\(username)\AppData\Roaming\Mozilla\Firefox" and select the folder (OK in the small window)

"Add" button: Navigate to "C:\Users\(username)\AppData\Local\Mozilla\Firefox " and select the folder.
OK your way out.

The sandbox is now ready for the install of Firefox.
Leave the above settings in your Sandbox Settings, even after the Firefox install is complete.

4. Right-click the Firefox installer program; choose "Run Sandboxed"; chose the sandbox to use; and let the install complete and allow Firefox to start. This will create all of the necessary files inside of the sandbox.
5. The next time you want to use Firefox, use the "Run From Start Menu" for the sandbox, and Firefox should be listed at the top of the "Desktop" listing.

Optional:
6. Add another setting to the sandbox, so you don't accidentally choose to Delete Contents:
Sandbox Settings > Delete > Delete Invocation > "Never remove this sandbox or delete it's contents"
7. You will probably want to remove your "Desktop" folder from Sandboxie's list of Recover Folders. Otherwise, the "Firefox" shortcut that appears at the top of the Sandboxie Start Menu > Desktop might be recovered to your "real" Desktop.
Sandbox Settings > Recovery > Quick Recovery > select the "Desktop" entry, and "Remove" it.

this is to specific and can be shorter if sandboxie can deal with system variables:

installation x86
%programfiles%\Mozilla

installation x64
%programfiles%\Mozilla <-- will do also if caller is 32 bit due wow6432node detour
%programfiles(x86)%\Mozilla

profiles
%appdata%\Mozilla\Firefox\profiles

cache
%localappdata%\Mozilla\Firefox\profiles

you can put those strings also into START > execute - explorer will open that path

http://de.wikipedia.org/wiki/Umgebungsvariable

did you use the complete setup file or the automatic update?
the auto update may fail cause firefox uses now a service and it may possible that sandboxie cant deal this.