I just barely learned that IMAP is a protocol that allows you to read and respond to emails without using the internet. Would it be possible to use this application for packet radio and if so, how? I am using the Ubuntu 14.10 operating system.
IMAP is a protocol for transferring email, nothing more, nothing less. It typically uses TCP/IP as the underlying protocol, but it could be routed over something else (like packet radio). If you wanted to, you could configure TCP/IP over AX.25 but I don't know if anyone else is doing it, so you'd probably need to run your own mailserver on the other end as well. (You could even configure that mailserver to bridge to the internet, but keep in mind that you'd be responsible for any incoming traffic that made it onto amateur frequencies!)
What you can't do is use Thunderbird to interact with any typical packet BBS or mailbox systems, they work with different protocols. So essentially, what you ask is possible, I'm just not sure if it's useful.
While I'm not very experienced with packet radio (it's on my list of projects), I can comment in general on using IMAP on low-bandwidth connections. I have extensive experience using it on dial-up TCP/IP connections (PPP), which are much slower than modern broadband connections and have much higher latency. These are aspects it has in common with packet radio connections, although packet radio is much slower still (I was getting about 40kbps down and 28kbps up on dialup, and packet is either 1200 or 9600 bps bidirectionally and single duplex).
IMAP is ... well ... nearly unbearable on low bandwidth and high latency connections. It takes very long periods of time to negotiate connections and download messages. Watching the traffic using a packet analyzer (tcpdump), the protocol is very chatty. With packet radio being single duplex, I think using IMAP over packet radio would verge on being unbearable unless the amount of email in the IMAP account is very small, indeed.
That having been said, by all means, experiment with it and post your findings. I may just try the same to satisfy my instincts about it.