I see your 386 and raise you an 8088 at 4.77 MHz and 256K of RAM (or less, I'm not sure what the actual requirements of ircjr is if you want to push it): http://www.brutman.com/mTCP/IRCjr.html
Really, it's just text, it doesn't matter. Literally any general purpose computer will run IRC if it can do TCP/IP and someone bothers to write a client.