I've found two applications that "do it":
mIRGGI: http://mirggi.net/
Before you download either one of them, here is a quick summary of my experiences.
mIRGGI:
Made for S60 2nd, 3rd and 5th edition (so it's not a Java app). Looks like a real IRC client, with chat window and a line for typing below it. You can customize colours. You can have several active server connections and channels. URLs posted on channels can be picked up from the list (latest first) and copied to system clipboard. Text from the channel can be selected and copied. Logging to text file is possible.
The downside: crashes occasionally, most likely when the mobile internet connection breaks. Also seems to crash when you go to settings while having open session. May run a whole day fine if the connection is stable. But this is clearly the biggest problem: the more you move, the more it crashes. Ironic, isn't it? You'd expect it to be a mobile application!
JMirc:
Java, should run on any J2ME device. This is a bit simpler application, but works very well. You can have multiple channels, but not on different servers. Text is entered using the full screen system editor (while you're typing, there is nothing else on the screen than what you are typing), and you cannot send a line by pressing enter. Those may be a "NO" to an old IRCer. If you're willing to stick to the subject, and don't have S60 phone, this might be it. I don't remember any crashes, though I'm not sure what happens when the connection breaks.
Others?
To my understanding, those are the first options on mobile IRCing. I don't have experience on server ircing (ssh or what ever connections, I've just heard about such option) or other client applications, but if YOU do, please leave a comment!
