Thunderbird handles offline messages in quite a particular way. Actually, it only downloads messages when you read them, and that, even if you select the “Download messages for offline use when going offline” option and see emails in your local folder. Consequently, messages for which you get the warning “The selected message is on an IMAP server that requires login” are probably those that are not read (those not really downloaded).
To avoid getting this warning you might want to do the following on a regular basis, such as every morning:
1. Select all unread messages.
2. Change their status from unread to read (this will download the messages).
3. Put the emails back to their unread status.
This simple procedure should allow your Desktop Search software to index and display the contents of offline messages. You should no longer receive the warning.