If you change your IP address on your client's socket to an "anonymous" IP address (one on a different physical subnet than your own assigned address), you will not be able to receive replies sent to that IP address unless you also manipulate the routing tables of all of the routers between the IRC server and your "anonymous" client. You probably can't do that, so it's not actually useful to know how.|
Note that if your machine is physically on an ethernet segment with a subnet, you could just change your machine's IP address to a different address within the same subnet, which would obscure your identity with that of another user on the same subnet (i.e. "they" will know what company or ISP you're from but not which particular user, unless they have some other information to identify you). Cable modems are good for this, as they often have little security or accounting and lots of spare addresses to choose from.
Kids, don't try this at home. People who can afford lawyers get seriously offended if you steal their vacant IP addresses.