This is all a storm in a teacup as far as I can tell. Besides being a MITM the attacker also needs to be able to control a message going out of either the client or server precisely and in its entirety. It's not enough to simply know what mIRC (or the server) is sending and when, it needs to be able to tell mIRC/server exactly what to send. As Sat said, in a protocol like IRC this just isn't feasible. The IRC server obviously requires that parts of the message be valid in both format and content that would be impossible for the attacker to maintain (we're talking about XOR'ing encrypted data with a regular IRC message and having the result still be a valid IRC message that the server will accept and deliver to the correct recipient [the attackee]).
In short neither the original TLS 1.0 attack nor the improved "BEAST" attack pose any legitimate threat to secure IRC connections.
Note that I'm saying all this on the basis of the attack as described in the link you provided. I haven't read any of the original papers.
