This article was written for Thunderbird but also applies to Mozilla Suite / SeaMonkey (though some menu sequences may differ). It hasn't been updated for Thunderbird 3 yet, which adds support for a different type of offline folder in Tools -> Account Settings -> Synchronization & Storage
Thunderbird normally downloads just the headers for messages in a IMAP account. It will fetch the contents of a message as needed, but it won't store it in the profile. There are several different ways to automate downloading messages to your hard disk so that they can be used when working offline. The most common way is to configure several folders for offline use and get prompted whether it should synchronize them when you tell Thunderbird you're going to work offline. However, if you don't want to automate downloading the messages you can get a snapshop of a folder by right clicking on the remote folder, selecting Properties, the Offline tab and pressing the "Download Now" button.
Note: Be advised that the initial synchronization may involve downloading a substantial number of messages large in size. Some providers may impose bandwidth limitations, and there are reported cases where an account was temporarily locked down because that limit was exceeded during offline synchronization. Synchronization should resume once the account is re-enabled.
Configure the folder for offline use
You can configure a folder for offline use with either Tools -> Account Settings -> Offline & Disk Space or by right clicking on the remote folder, selecting Properties, the Offline tab and then checking the "Select this folder for offline use" checkbox. The downloaded folder is only accessible when you've switched to working offline using File -> Offline -> Work Offline. Its not effected by whether or not your PC is offline.
Both Tools -> Options -> Advanced -> Network & Disk Space -> Offline and File -> Offline -> Offline Settings let you configure Thunderbird to automatically download/sync when you go offline. You could also tell it to download the offline folders using File -> Offline -> Download/Sync Now.
The MIME Parts On Demand preferences control how much of a message is fetched when you read it. If you want the attachments downloaded with the message set mail.server.default.mime_parts_on_demand to false (that effects all IMAP accounts). You can use mail.server.serverX.mime_parts_on_demand instead if you want to specify it per account.
A sanity check that it actually downloaded the messages is to see if it created a mbox file for that folder in the appropriate account directory in your profile. The mbox file for your inbox folder is called "inbox.". The "inbox.msf" file is the headers while Inbox.sbd is a directory.
If Thunderbird doesn't recognize that a folder contains unread messages try setting mail.imap.use_status_for_biff false. It causes Thunderbird to explicitly select each folder to update the message summaries, rather than use the STATUS command. It has more overhead, but some IMAP servers don't return STATUS correctly.
Sync on Arrival
This add-on will be obsolete when Thunderbird 3 is released.
The Sync on Arrival extension can also be used to download messages when Thunderbird detects new messages in a folder. It requires both "Check this folder for new messages" and "Select this folder for offline use" be enabled. The extension also keeps any saved search folders (for IMAP accounts) up to date even if they're configured for offline use.
The author's description talks about it implementing what the download_on_biff preference promises but fails to deliver but its not clear why that statement wouldn't also apply to the built-in offline folder support. It has a "Process message download queue one message at a time" option as a workaround if only the first of several messages is downloaded. That sounds useful, but there haven't been any reports in the forums of that being a problem with the built-in offline folder support. You need to bypass the version check per this article since it only supports version 1.0 – 1.5.0.*. There is a dedicated thread for it here.
Thunderbird provides a mechanism for message aging to automatically delete "old" messages. A retention period and other policies can be defined in the Offline & Disk Space settings of an account or the Folder Properties of specific folders. This applies not just apply to the local offline copies, but also to the remote copy on the server as well. Any messages deleted via this feature are permanently deleted, not moved to the Trash folder. Its a good idea to check the message aging settings to avoid accidental data loss due to this feature.
You can force Thunderbird to download messages for offline use whenever you open a folder or Thunderbird detects new mail by setting mail.server.default.autosync_offline_stores to true. You can use mail.server.serverX.autosync_offline_stores instead if you want to specify it per account. This only works if you also set mail.imap.use_status_for_biff false. It ignores whether the folder is configured as a offline folder or not.   
The advantage of the AutoSync settings is that its more transparent, and any downloading is not delayed until you go offline.
This feature doesn't appear to be widely used (yet). If you use it be careful about how you configure Thunderbird to check for new mail to avoid it downloading folders that you didn't expect. For example, it might download messages for a folder that was unsubscribed, due to the IMAP server sending a IDLE command to notify Thunderbird that there were new messages in that folder.
None of the autosync_offline_stores preferences exist, though Thunderbird has built-in support for them. You need to create them. The easiest way is to use the Config Editor. See this article for how to do that.
Use message filters
If you get all of your new mail in the inbox its much simpler to just use a message filter to automatically copy or move any new messages to a folder in the local folders directory. The main problem with using message filters is they are only run automatically for the inbox, you have to use "Tools -> Run Filters on Folder" for any other folder.
Save it as a .eml file
If you save a message as a .eml file you can view it by double clicking on it in Windows Explorer, but you can't import it as a real message in Thunderbird. The ImportExportTools extension lets you select several .eml files and import them as real messages.
The Autosave extension can automatically save any new messages in the specified folders as .eml files. It does not have a GUI, this forum thread describes how to configure it using preferences in prefs.js. One problem with using this extension is that the author stopped supporting their extensions years ago and has not been heard from since. You also need to bypass the version check per this article.
- Checking for new messages in other folders
- Deleting messages in IMAP accounts
- Entire message fetched when opening a IMAP message
- Modify Thunderbird settings
- Bug report about adding the ability to auto sync IMAP folders offline store when folders are opened or checked for new mail.
- Bug report about automatically downloading full messages on IMAP folders.
- Bug report about download IMAP messages for offline use immediately
- Bug report about the integration of Thunderbird with Vista Desktop Search. One of the developers advocates enabling autosync by default when they do that.
- Bug report about "-offline" command line argument doesn't work under Windows.
- IMAPsize is a Windows utility that can do a incremental backup of remote folders to .eml files.
- Offline extension spec. That extension is now built-in.