A 386 w/5mb of ram. Or at least it was when I was doing so in 1995.
https://en.wikipedia.org/wiki/MIRC
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.
Most ISA network cards are for the 16-bit bus, XT class machines only have 8-bit slots. This doesn't necessarily mean that the cards will not function, but you also need a packet driver and they very often use 186/286 instructions and will not run at all on an 8088.
The Intel EtherExpress 8∕16 will work, so will the 3Com 3C501. You may also have some luck with generic NE1000/NE2000 clones.